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Statistics 5802 Game Theory Overview of games 2 player games representations 2 player zero-sum games Render/Stair/Hanna text CD QM for Windows software Modeling Lecture Agenda A model of reality Elements Players Rules Strategies Payoffs What is a game?

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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
Lecture Agenda
what is a game

A model of reality

  • Elements
    • Players
    • Rules
    • Strategies
    • Payoffs
What is a game?
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

Players
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?
Rules
strategies

Strategies - a complete specification of what to do in all situations

strategy versus move

Examples –

tic tac toe; let\'s make a deal

Strategies
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)
Payoffs
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
Game classifications
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)
Goals when studying games
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.

Example: Battle of the sexes
game issues
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.

the general 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
the general two player zero sum game20

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
The General Two Player Zero Sum Game
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
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

2 by 2 Sample Row, Column Interchange
example 1 row s choice
Example 1 - Row’s 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 column s choice
Example 1 – Column’s 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
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
Using Domination

We can always eliminate rows or columns which are dominated in a zero sum game.

using domination28
Using Domination

We can always eliminate rows or columns which are dominated in a zero sum game.

example 1 game solution
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.

a notion of fair
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
example 2
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 row s choice
Example 2 - Row’s 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 column s choice
Example 2 - Column’s 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 < $24

But $55 > $30

Therefore, neither column dominates the other.

simple games 2 column s choice continued
Simple games - #2Column’s choice – continued

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
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 336
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 row s conservative approach
Example 3 - Row’s 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.

example 3 maximin
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 column s conservative way
Example 3 - Column’s 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.

example 3 minimax
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
Example 3 - Solution ???

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 solution42
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 solution43
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 solution44
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 solution45
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 solution46
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 solution47
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.

the princess bride
The Princess Bride

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

examination of game 3

Notice that in game 3 (which is hard to solve) we have that

maximin < minimax. The Value of the game is between maximin, minimax

Examination of game 3

Minimax

maximin

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
Mixed strategies
expected values weighted average as a function of p
Expected values (weighted average) as a function of p

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

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
Solution
example results
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.

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

ExpectED value computation
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
Solution summary
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.

Zero-sum Game Features
models see word document
Models

(see Word document)

Game Theory

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