# MATH 1007 Chapter Notes - Chapter 13-17: North South Mrt Line, Best Response, Saddle Point

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Samuel Mohebban Exam 3 Notes Math and Politics

Chapter 13- Strategies and Outcomes

12

5

10

6

-13

-7

There are two players in this game, alled ro ad olu (oth players ko all

the entries in the matrix)

Row secretly chooses a row (2 choices available to him)

Column secretly chooses a column (3 choices available to him)

The two players then reveal their choices simultaneously

EX. If row chooses row 2 and column hooses olu , the etry 6 i the loer left is

identified. This entry represents the number of dollars that column must pay to row

o If the entry were negative, such as -12, then row would have to pay column

o If row and column both select their first option, then row must pay column

Two-person sum Game

o One person always pays the other

o Rock Paper Scissors

Two people simultaneously show their hands, options being either rock,

paper, or scissor.

The choices in the game are called the players strategies

The two players are called row who is female and column who is male

The strategy chosen by the two players is called the outcome

o In some other cases where the outcome is more complicated then just winning

or losing, we institute a score-keeping system that awards a numerical score to

each player after each round is played.

Because this score is frequently a monetary payment, it is called a payoff

In rock paper scissors, we may require the loser to pay $1 to the winner

or $0 if it is a tie

The ier payoff is $ ad the loser’s payoff is -$1

The su of the loser’s payoff ad the ier’s payoff is alays $

This is why rock paper scissors is a zero-sum game

Zero-sum games

o Row has m strategy choices

o Column has n strategy choices

o This creates a game by an m-by-n matrix in which the rows correspond to Row’s

strategies and the columns corresponds to colu’s strategies

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Samuel Mohebban Exam 3 Notes Math and Politics

o Involves a transfer of wealth between two players; no new wealth is gained or

lost

o The sign of ui,j indicates which player pays and which player gets paid

o m-by-n matrix

An array of number with m rows and n columns

The matrix entry that lies in row i and column j,

which we donate ui,j, is the payoff to row for the outcome (row i, column

j), which is called the payoff matrix

The payoff matrix is called matrix game

If the payoff to row for the outcome (row i, column j) is given by ui,j, then

the payoff to column for the same outcome is - ui,j

Because the sum is always 0

ui,j + (-ui,j) = 0 for all i and j

Rock

Paper

Scissors

Rock

0

-1

1

Paper

1

0

-1

Scissors

-1

1

0

In this matrix the entry u2,1 =1

o Indicates that if row plays row 2 (paper), and column plays column 1 (rock), then

row must pay column $1

Guarantee of a strategy

o A play can be secure in the knowledge that the consequences of a certain

strategy choice can be no worse than its guarantee

o For row, the guarantee is the smallest entry in the row

o For column, the guarantee is largest entry in the column

Prudent Method- when a player chooses the strategy with the best guarantee

o We donate r for row’s prudent strategy

o We donate c for olu’s prudet strategy

o Uses the orst ase aalysis

One imagines the worst possible result for each strategy choice, and

selects the one whose worst case is the least bad

o Also known as the pessimistic strategy

o EX.

For the Americans, the guarantee for the strategy North (row 1) is 2 days

of bombing, and the guarantee for South (row 2) is 1 day of bombing.

Since 2 days of bombing is better than 1 day, North is the American

prudent strategy; r=2

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