Game Theory

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Game Theory

• Statistics 5802

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Lecture Agenda

• Overview of games
• 2 player games
• representations
• 2 player zero-sum games
• Render/Stair/Hanna text CD
• QM for Windows software
• Modeling

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What is a game?

• A model of reality
• Elements
• Players
• Rules
• Strategies
• Payoffs

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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

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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. lets make a deal
• Are moves sequential or simultaneous?

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Strategies

• Strategies - a complete specification of what to
  do in all situations
• strategy versus move
• Examples
• tic tac toe let's make a deal

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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 (lets 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)

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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

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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)

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2 player game representations

• Table generally for simultaneous moves
• Tree generally for sequential moves

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Example Battle of the sexes

• A woman (Ellen) and her husband (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.

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Payoff Table
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Game issues
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.
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Game tree example Ellen goes first
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Game tree solution - solve backwards (right to
left)

• Determine what Pat would do at each of the Pat
  nodes

Compare 1 and -1
Compare -1 and 2
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Game tree solution - solve backwards (right to
left)

• then determine what Ellen should do

Compare 1 and -1
Compare 2 and 1
Compare -1 and 2
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Observation

• In a game such as the Battle of the Sexes a
  preemptive decision will win the game for you!!

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The 2 player zero sum game
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The General Two Player, Zero Sum Game

• 2 players
• Opposite interests (zero sum)
• communication does not matter
• binding agreements do not make sense

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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

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Sample Game Matrix

• Column pays row the amount in the cell
• Negative numbers mean row pays column

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2 by 2 Sample

• 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

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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

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Example 1 - Rows choice
Reminder Column pays row the amount in the
chosen cell.
You are row. Should you select row 1 or row 2 and
why? Remember, row and column select
simultaneously.
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Example 1 Columns choice
Reminder Column pays row the amount in the
chosen cell.
You are column. Should you select col 1 or col 2
and why? Remember, row and column select
simultaneously.
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Domination
Reminder Column pays row the amount in the
chosen cell.
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.
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Using Domination
We can always eliminate rows or columns which are
dominated in a zero sum game.
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Using Domination
We can always eliminate rows or columns which are
dominated in a zero sum game.
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Example 1 - Game SolUTION
Reminder Column pays row the amount in the
chosen cell.
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.
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A Notion of Fair

• Game
• Splitting a piece of cake
• In two
• Statistician
• Game theorist
• In more than two
• Team work division
• Splitting work for projects

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Example 2

• 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)

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Example 2 - Rows choice
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.
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Example 2 - Columns choice
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 lt 24 But 55 gt 30 Therefore,
neither column dominates the other.
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Simple games - 2Columns choice continued
However, when column examines this game, column
knows that row is going to select row 2.
Therefore, columns 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.
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Example 3
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)
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Example 3
This game has no dominant row nor does it have a
dominant column. Thus, we have no straightforward
answer to this problem.
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Example 3 - Rows conservative approach
Row could take the following conservative
(maximin) 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.
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Example 3 - Maximin
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.
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Example 3 - Columns conservative way
Column could take a similar conservative
(minimax) 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.
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Example 3 - Minimax
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.
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Example 3 - Solution ???
When we put row and columns 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?
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Example 3 - Solution ???
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.
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Example 3 - Solution ???
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.
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Example 3 - Solution ???
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.
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Example 3 - Solution ???
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.
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Example 3 - Solution ???
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 .
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Example 3 - Solution ???
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.
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Optimal strategy

• You must select your strategy randomly!!!

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The Princess Bride

• http//www.imdb.com/title/tt0093779/

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Examination of game 1
Minimax
maximin

• Notice that in examples 1 2 (which are trivial
  to solve) we have that
• maximin minimax

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Examination of game 3
Minimax
maximin

• Notice that in game 3 (which is hard to solve) we
  have that
• maximin lt minimax. The Value of the game is
  between maximin, minimax

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Mixed strategies

• 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

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Expected values (weighted average) as a function
of p
How will column respond to any value of p for row?
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Graph of expected value as a function of rows mix
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Solution

• 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

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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.
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ExpectED value computation

• 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

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ExpectED value computation (continued)

• This leads to an expected value of
• 25.27667.04734.57914.098 31.097

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Solution summary

• If maximinminimax
• there is a saddle point (equilibrium) and each
  player has a pure strategy plays only one
  strategy
• If maximin does not equal minimax
• maximin lt value of game lt minimax
• We find mixed strategies
• We find the (expected) value or weighted average
  of the game

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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.

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Game Theory

• Models
• (see Word document)