game theory statistics 802. lecture agenda overview of games 2 player games representations 2 player...
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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 naturedecision 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 Pat2 , 1
WWEPat
Ballet-1 , -1
Ellen
WWE-1 , -1
BalletPat
Ballet1 , 2
Game tree solution - solve backwards (right to left)
Determine what Pat would do at each of the Pat nodes …
PayoffsWWE Ellen Pat
2 , 1WWE 2,1
Ballet-1 , -1
Ellen
WWE-1 , -1
Ballet 1, 2
Ballet1 , 2
Pat
Pat
Compare 1 and -1
Compare -1 and 2
Game tree solution - solve backwards (right to left)
… then determine what Ellen should do
PayoffsWWE Ellen Pat
2 , 1WWE 2,1
Ballet-1 , -1
WWE-1 , -1
Ballet 1, 2
Ballet1 , 2
Pat
Pat
Compare 1 and -1
Compare -1 and 2
Compare 2 and 1
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 . 45Row Strat 2 -54 22 . -67 . . . .
Row Strat m 73 54 . 52
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
col 1 col 2row 1 25 67row 2 34 14
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!!
col 1 col 2row 1 25 67row 2 34 14
col 1 col 2row 2 34 14row 1 25 67
Example 1 - Row’s choiceReminder: 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.
col 1 col 2row 1 $11 $27row 2 $34 $42
Example 1 – Column’s choiceReminder: 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.
col 1 col 2row 1 $11 $27row 2 $34 $42
DominationReminder: 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.
col 1 col 2row 1 $11 $27row 2 $34 $42
Using Domination
We can always eliminate rows or columns which are dominated in a zero sum game.
col 1 col 2row 1 $11 $27row 2 $34 $42
Using Domination
We can always eliminate rows or columns which are dominated in a zero sum game.
Example 1 - Game SolutionReminder: 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.
col 1 col 2row 1 $11 $27row 2 $34 $42
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)
col 1 col 2row 1 $18 $24row 2 $55 $30
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.
col 1 col 2row 1 $18 $24row 2 $55 $30
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.
col 1 col 2row 1 $18 $24row 2 $55 $30
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.
col 1 col 2row 1 $18 $24row 2 $55 $30
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)
col 1 col 2row 1 25 67row 2 34 14
Example 3
This game has no dominant row nor does it have a dominant column. Thus, we have no straightforward answer to this problem.
col 1 col 2row 1 25 67row 2 34 14
Example 3 - Row’s 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.
col 1 col 2 worstrow 1 25 67 25row 2 34 14 14
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.
col 1 col 2 worstrow 1 25 67 25row 2 34 14 14
Example 3 - Column’s 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.
col 1 col 2row 1 $25 $67row 2 $34 $14worst $34 $67
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.
col 1 col 2row 1 $25 $67row 2 $34 $14worst $34 $67
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?
col 1 col 2 worstrow 1 $25 $67 $25row 2 $34 $14 $14worst $34 $67
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.
col 1 col 2row 1 $25 $67row 2 $34 $14
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.
col 1 col 2row 1 $25 $67row 2 $34 $14
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.
col 1 col 2row 1 $25 $67row 2 $34 $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.
col 1 col 2row 1 $25 $67row 2 $34 $14
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 …
.
col 1 col 2row 1 $25 $67row 2 $34 $14
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.
col 1 col 2row 1 $25 $67row 2 $34 $14
Optimal strategy
You must select your strategy randomly!!!
Examination of game 1
col 1 col 2worst (row minimum)
row 1 $11 $27 $11row 2 $34 $42 $34worst (column maximum) $34 $42
Notice that in examples 1 & 2 (which are trivial to solve) we have that
maximin = minimax
maximinMinimax
col 1 col 2worst (row minimum)
row 1 $25 $67 $25row 2 $34 $14 $14worst (column maximum) $34 $67
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
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
q 1-qcol 1 col 2
row 1 p 25 67row 2 1-p 34 14
Expected values (weighted average) as a function of p
col 1 col 2row 1 p 25 67row 2 1-p 34 14
p vs. col 1 vs col 20 34 14
0.1 33.1 19.30.2 32.2 24.60.3 31.3 29.90.4 30.4 35.20.5 29.5 40.50.6 28.6 45.80.7 27.7 51.10.8 26.8 56.40.9 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
01020
30405060
7080
0 0.2 0.4 0.6 0.8 1
p
Pla
yer
1's
pa
yoff
vs. col 1 vs col 2
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
col 1 col 2row 1 25 67row 2 34 14
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
Col strat 1 Col strat 2Row strat 1 25 67Row strat 2 34 14
Col strat 1 Col strat 2 probabilitiesRow strat 1 0.275754 0.046826 0.322581Row strat 2 0.579084 0.098335 0.677419probabilities 0.854839 0.145161
Expect value computation (continued)
This leads to an expected value of25*.276+67*.047+34*.579+14*.098 = 31.097
Col strat 1 Col strat 2Row strat 1 25 67Row strat 2 34 14
Col strat 1 Col strat 2 probabilitiesRow strat 1 0.275754 0.046826 0.322581Row strat 2 0.579084 0.098335 0.677419probabilities 0.854839 0.145161
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 0A graph can be drawn for a player if the
player has only 2 strategies available.
Game Theory
Models(see Word document)