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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. lets 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 (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)

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

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.

Game tree example Ellen goes first

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

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

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

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

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!!

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.

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.

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.

Using Domination

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.

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

- 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 - 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.

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.

Simple games - 2 Columns 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.

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)

Example 3

This game has no dominant row nor does it have a

dominant column. Thus, we have no straightforward

answer to this problem.

Example 3 - Rows conservative approach

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

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 - Columns conservative way

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

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 ???

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?

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.

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.

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.

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.

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 .

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.

Optimal strategy

- You must select your strategy randomly!!!

The Princess Bride

- http//www.imdb.com/title/tt0093779/

Examination of game 1

Minimax

maximin

- Notice that in examples 1 2 (which are trivial

to solve) we have that - maximin minimax

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

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

Expected values (weighted average) as a function

of p

How will column respond to any value of p for row?

Graph of expected value as a function of rows mix

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

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

- 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)

- This leads to an expected value of
- 25.27667.04734.57914.098 31.097

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

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)