Game Theory
Statistics 802
Lecture Agenda
• Overview of games
• 2 player games
• representations
• 2 player zero-sum games
• Render/Stair/Hanna text CD
• QM for Windows software
• Modeling
What is a game?
• A model of reality
• Elements
• Players
• Rules
• Strategies
• Payoffs
Players
Players - each player is an individual or group
of individuals with similar interests
(corporation, nation, team)
Single player game – game against nature
decision table
Rules
• To what extent can the players
communicate with one another?
• Can the players enter into binding
agreements?
• Can rewards be shared?
• What information is available to each
player?
• Tic-tac-toe vs. let’s make a deal
• Are moves sequential or simultaneous?
Strategies
Strategies - a complete specification of what to
do in all situations
strategy versus move
Examples –
tic tac toe; let's make a deal
Payoffs
• Causal relationships - players' strategies lead to
outcomes/payoffs
• Outcomes are based on strategies of all players
• Outcomes are typically $ or utils
• long run
• Payoff sums
• 0 (poker, tic-tac-toe, market share change)
• Constant (total market share)
• General (let’s make a deal)
• Payoff representation
• For many games if there are n-players the outcome is
represented by a list of n payoffs.
• Example – market share of 4 competing companies -
(23,52,8,7)
Game classifications
• Number of players
• 1, 2 or more than 2
• Total reward
• zero sum or constant sum vs non zero sum
• Information
• perfect information (everything known to
every player) or not
• chess and checkers - games of perfect information
• bridge, poker - not games of perfect information
Goals when studying games
• Is there a "solution" to the game?
• Does the concept of a solution exist?
• Is the concept of a solution unique?
• What should each player do? (What are the optimal
strategies?)
• What should be the outcome of the game? (e.g.-tic tac
toe – tie; )
• What is the power of each player? (stock holders,
states, voting blocs)
• What do (not should) people do (experimental,
behavioral)
2 player game
representations
• Table – generally for simultaneous moves
• Tree – generally for sequential moves
Example: Battle of the sexes
A woman (Ellen) and her partner (Pat)
each have two choices for entertainment
on a particular Saturday night. Each can
either go to a WWE match or to a ballet.
Ellen prefers the WWE match while Pat
prefers the ballet. However, to both it is
more important that they go out together
than that they see the preferred
entertainment.
Payoff Table
Ellen\Pat WWE Ballet
WWE (2, 1) (-1, -1)
Ballet (-1, -1) (1, 2)
Game issues
Ellen\Pat WWE Ballet
WWE (2, 1) (-1, -1)
Ballet (-1, -1) (1, 2)
Do players see the same reward structure? (assume yes)
Are decisions made simultaneously or does one player go first?
(If one player goes first a tree is a better representation)
Is communication permitted?
Is game played once, repeated a known number of times or
repeated an “infinite” number of times.
Game tree example – Ellen
goes first
WWE Ellen Pat
2, 1
WWE
Pat
Ballet
-1 , -1
Ellen
WWE
-1 , -1
Ballet
Pat
Ballet
1, 2
Game tree solution - solve
backwards (right to left)
Determine what Pat would do at each of
the Pat nodes …
Payoffs
WWE Ellen Pat
2, 1
WWE 2,1
Pat Compare 1
Ballet
and -1
-1 , -1
Ellen
WWE
-1 , -1
Ballet 1, 2 Compare -
Pat
1 and 2
Ballet
1, 2
Game tree solution - solve
backwards (right to left)
… then determine what Ellen should do
Payoffs
WWE Ellen Pat
2, 1
WWE 2,1
Pat Compare 1
Ballet
and -1
Compare 2 -1 , -1
and 1
WWE
-1 , -1
Ballet 1, 2 Compare -
Pat
1 and 2
Ballet
1, 2
Observation
• In a game such as the Battle of the Sexes
a preemptive decision will win the game
for you!!
The 2 player zero sum game
The General (m by n) Two
Player, Zero Sum Game
• 2 players
• opposite interests (zero sum)
• communication does not matter
• binding agreements do not make sense
The General Two Player
Zero Sum Game
• Row has m strategies
• Column has n strategies
• Row and column select a strategy
simultaneously
• The outcome (payoff to each player) is a
function of the strategy selected by row and the
strategy by column
• The sum of the payoffs is zero
Sample Game Matrix
• Column pays row the amount in the cell
• Negative numbers mean row pays column
Col 1 Col 2 Col n
Row Strat 1 20 -35 . 45
Row Strat 2 -54 22 . -67
. . . .
Row Strat m 73 54 . 52
2 by 2 Sample
c ol 1 c ol 2
ro w 1 25 67
ro w 2 34 14
• Row collects some amount between 14 and 67
from column in this game
• Decisions are simultaneous
• Note: The game is unfair because column can
not win. Ultimately, we want to find out exactly
how unfair this game is
2 by 2 Sample
Row, Column Interchange
• Rows, columns or both can be interchanged
without changing the structure of the game. In
the two games below Rows 1 and 2 have been
interchanged but the games are identical!!
c ol 1 c ol 2
row 1 25 67
row 2 34 14
c ol 1 c ol 2
row 2 34 14
row 1 25 67
Example 1 - Row’s choice
Reminder: Column pays row the amount in the chosen cell.
c ol 1 c ol 2
ro w 1 $11 $27
ro w 2 $34 $42
You are row. Should you select row 1 or row 2 and why?
Remember, row and column select simultaneously.
Example 1 – Column’s choice
Reminder: Column pays row the amount in the chosen cell.
c ol 1 c ol 2
ro w 1 $11 $27
ro w 2 $34 $42
You are column. Should you select col 1 or col 2 and why?
Remember, row and column select simultaneously.
Domination
Reminder: Column pays row the amount in the chosen cell.
c ol 1 c ol 2
ro w 1 $11 $27
ro w 2 $34 $42
We say that row 2 dominates row 1 since each outcome in row
2 is better than the corresponding outcome in row 1
Similarly, we say that column 1 dominates column 2 since each
outcome in column 1 is better than the corresponding outcome
in column 2.
Using Domination
c ol 1 c ol 2
ro w 1 $11 $27
ro w 2 $34 $42
We can always eliminate rows or columns which are
dominated in a zero sum game.
Using Domination
We can always
eliminate rows or
columns which are
dominated in a zero
sum game.
Example 1 - Game Solution
Reminder: Column pays row the amount in the chosen cell.
c ol 1 c ol 2
ro w 1 $11 $27
ro w 2 $34 $42
Thus, we have solved our first game (and without using QM
for Windows.) Row will select row 2, Column will select col 1
and column will pay row $34. We say the value of the game is
$34. We previously had said that this game is unfair because
row always wins. To make the game fair, row should pay
column $34 for the opportunity to play this game.
Example 2
c ol 1 c ol 2
ro w 1 $18 $24
ro w 2 $55 $30
Answer the following 3 questions before going to the
following slides.
•What should row do? (easy question)
•What should column do? (not quite as easy)
•What is the value of the game (easy if you got the other 2
questions)
Example 2 - Row’s choice
c ol 1 c ol 2
ro w 1 $18 $24
ro w 2 $55 $30
As was the case before, row should select row 2 because it is
better than row 1 regardless of which column is chosen. That
is, $55 is better than $18 and $30 is better than $24.
Example 2 - Column’s choice
c ol 1 c ol 2
ro w 1 $18 $24
ro w 2 $55 $30
Until now, we have found that one row or one column
dominates another. At this point though we have a problem
because there is no column domination.
$18 $30
Therefore, neither column dominates the other.
Simple games - #2
Column’s choice – continued
c ol 1 c ol 2
ro w 1 $18 $24
ro w 2 $55 $30
However, when column examines this game, column knows
that row is going to select row 2. Therefore, column’s only
real choice is between paying $55 and paying $30. Column
will select col 2, and lose $30 to row in this game.
Notice the “you know, I know” logic.
Example 3
col 1 col 2
row 1 25 67
row 2 34 14
Answer the following 3 questions before going to the
following slides.
What should row do? (difficult question)
What should column do? (difficult question)
What is the value of the game (doubly difficult question since
the first two questions are difficult)
Example 3
col 1 col 2
row 1 25 67
row 2 34 14
This game has no dominant row nor does it have a dominant
column. Thus, we have no straightforward answer to this
problem.
Example 3 -
Row’s conservative approach
col 1 col 2 worst
row 1 25 67 25
row 2 34 14 14
Row could take the following conservative approach to this
problem. Row could look at the worst that can happen in
either row. That is, if row selects row 1, row may end up
winning only $25 whereas if row selects row 2 row may end
up winning only $14. Therefore, row prefers row 1 because
the worst case ($25) is better than the worst case ($14) for
row 2.
Example 3 - Maximin
col 1 col 2 worst
row 1 25 67 25
row 2 34 14 14
Since $25 is the best of the worst or maximum of the minima
it is called the maximin.
This is the same analysis as if row goes first.
Note: It is disadvantageous to go first in a zero sum game.
Example 3 -
Column’s conservative way
col 1 col 2
row 1 $25 $67
row 2 $34 $14
worst $34 $67
Column could take a similar conservative approach. Column
could look at the worst that can happen in either column. That
is, if column selects col 1, column may end up paying as
much as $34 whereas if column selects col 2 column may end
up paying as much as $67. Therefore, column prefers col 1
because the worst case ($34) is better than the worst case
($67) for column 2.
Example 3 - Minimax
col 1 col 2
row 1 $25 $67
row 2 $34 $14
worst $34 $67
Since $34 is the best of the worst or minimum of the maxima
for column it is called the minimax.
This is the same analysis as if column goes first.
Note: It is disadvantageous to go first in a zero sum game.
Example 3 - Solution ???
col 1 col 2 worst
row 1 $25 $67 $25
row 2 $34 $14 $14
worst $34 $67
When we put row and column’s conservative approaches
together we see that row will play row 1, column will play
column 1 and the outcome (value) of the game will be that
column will pay row $25 (the outcome in row 1, column 1).
What is wrong with this outcome?
Example 3 - Solution ???
col 1 col 2
row 1 $25 $67
row 2 $34 $14
What is wrong with this outcome?
If row knows that column will select column 1 because
column is conservative then row needs to select row 2 and get
$34 instead of $25.
Example 3 - Solution ???
col 1 col 2
row 1 $25 $67
row 2 $34 $14
However, if column knows that row will select row 2 because
row knows that column is conservative then column needs to
select col 2 and pay only $14 instead of $34.
Example 3 - Solution ???
col 1 col 2
row 1 $25 $67
row 2 $34 $14
However, if row knows that column knows that row will
select row 2 because row knows that column is conservative
and therefore column needs to select col 2 then row must
select row 1 and collect $67 instead of $14.
Example 3 - Solution ???
col 1 col 2
row 1 $25 $67
row 2 $34 $14
However, if column knows that row knows that column
knows that row will select row 2 because row knows that
column is conservative and therefore column needs to select
col 2 and that therefore row must select row 1 then column
must select col 1 and pay $25 instead of $67 and we are back
where we began.
Example 3 - Solution ???
col 1 col 2
row 1 $25 $67
row 2 $34 $14
The structure of this game is different from the structure of
the first two examples. They each had only one entry as a
solution and in this game we keep cycling around. There is a
lesson for this game …
.
Example 3 - Solution ???
col 1 col 2
row 1 $25 $67
row 2 $34 $14
The only way to not let your opponent take advantage of your
choice is to not know what your choice is yourself!!!
That is, you must select your strategy randomly. We call this
a mixed strategy.
Optimal strategy
You must select
your strategy
randomly!!!
The Princess Bride
http://www.imdb.com/title/tt0093779/
Examination of game 1
worst (row
col 1 col 2 minimum)
row 1 $11 $27 $11
row 2 $34 $42 $34
worst
(column Minimax
maximin
maximum) $34 $42
Notice that in examples 1 & 2 (which are trivial to
solve) we have that
maximin = minimax
Examination of game 3
worst (row
col 1 col 2 minimum)
row 1 $25 $67 $25
row 2 $34 $14 $14
worst Minimax
(column
maximin
maximum) $34 $67
Notice that in game 3 (which is hard to solve) we
have that
maximin < minimax. The Value of the game is
between maximin, minimax
Mixed strategies
q 1-q
col 1 col 2
row 1 p 25 67
row 2 1-p 34 14
• Row will pick row 1 with probability p and
row 2 with probability (1-p)
• For now, ignore the fact that column also
should mix strategies
Expected values (weighted
average) as a function of p
col 1 col 2
row 1 p 25 67
row 2 1-p 34 14
p vs. col 1 vs col 2
0 34 14
0.1 33.1 19.3
0.2 32.2 24.6
0.3 31.3 29.9
0.4 30.4 35.2
0.5 29.5 40.5
0.6 28.6 45.8
0.7 27.7 51.1
0.8 26.8 56.4
0.9 25.9 61.7
1 25 67
How will column respond to any value of p for row?
Graph of expected value
as a function of row’s mix
Example
80
70
Player 1's payoff
60
50
40
30
20
10
0
0 0.2 0.4 0.6 0.8 1
p
vs. col 1 vs col 2
Solution
col 1 col 2
row 1 25 67
row 2 34 14
• We need to find p to maximize the minimum
expected value against every column
• We need to find q to minimize the maximum
expected value against every row
Example - Results
Row should play row 1 32% of the time and row 2 68% of the
time. Column should play column 1 85% of the time and
column 2 15% of the time. On average, column will pay row
$31.10.
Expect value computation
Col strat 1 Col strat 2
Row strat 1 25 67
Row strat 2 34 14
Col strat 1 Col strat 2 probabilities
Row strat 1 0.275754 0.046826 0.322581
Row strat 2 0.579084 0.098335 0.677419
probabilities 0.854839 0.145161
If row and column each play according to the
percentages on the outside then each of the four cells
will occur with probabilities as shown in the table
Expect value computation
(continued)
Col strat 1 Col strat 2
Row strat 1 25 67
Row strat 2 34 14
Col strat 1 Col strat 2 probabilities
Row strat 1 0.275754 0.046826 0.322581
Row strat 2 0.579084 0.098335 0.677419
probabilities 0.854839 0.145161
This leads to an expected value of
25*.276+67*.047+34*.579+14*.098 = 31.097
Solution summary
• If maximin=minimax
• there is a saddle point (equilibrium) and each
player has a pure strategy – plays only one
strategy
• If maximin does not equal minimax
• maximin <= value of game <= minimax
• We find mixed strategies
• We find the (expected) value or weighted
average of the game
Zero-sum Game Features
A constant can be added to a zero sum game
without affecting the optimal strategies.
A zero sum game can be multiplied by a
positive constant without affecting the optimal
strategies.
A zero sum game is fair if its value is 0
A graph can be drawn for a player if the player
has only 2 strategies available.
Game Theory
Models
(see Word document)