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
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There are two players in this game, alled ro ad olu (oth players ko 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 etry 6 i the loer 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 ier payoff is $ ad the loser’s payoff is -$1
The su of the loser’s payoff ad the ier’s payoff is alays $
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 prudet strategy
o Uses the orst ase aalysis
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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