Statistics 5802

• Overview of games• 2 player games

• representations• 2 player zero-sum games

• Render/Stair/Hanna text CD• QM for Windows software

• Modeling

• A model of reality

• Elements• Players• Rules • Strategies• Payoffs

Players - each player is an individual or group of individuals with similar interests (corporation, nation, team)

Single player game – game against naturedecision table

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

strategy versus move Examples –

tic tac toe; let's make a deal

• 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)

• 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

• 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)

• Table – generally for simultaneous moves

• Tree – generally for sequential moves

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.

Ellen\Pat WWE Ballet

WWE (2, 1) (-1, -1)

Ballet (-1, -1) (1, 2)

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.

WWE Ellen Pat2 , 1

WWEPat

Ballet-1 , -1

Ellen

WWE-1 , -1

BalletPat

Ballet1 , 2

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

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

• New Yorker, The Back Page Royal Wedding Day Schedule for Her Majes

ty Queen Elizabeth

by Paul Rudnick March 14, 2005

7:00 a.m. Arise and greet corgis. Tell them it’s a very special day: they’ll be trying new Iams Lamb & Wild Rice in Gravy.

7:30 a.m. Breakfast with Prince Philip. Remind him that, even if they offer him a Jaguar, Al Qaeda are not friends.

8:00 a.m. Pick Corgi of the Day. Comfort and encourage remaining corgis. 8:30 a.m. Phone Fergie. Get her to send Camilla Weight Watchers frozen lasagna as gag

wedding gift. 9:00 a.m. Answer correspondence. Send Charles and Camilla a five-pound note as a wedding

gift, with card reading, “So sorry can’t attend ceremony in person. Hope face on this will suffice.”

9:30 a.m. Gather corgis to watch TiVo of “Desperate Housewives.” Discuss how much more romantic show would be if all characters were corgis.

10:00 a.m. Open local hospital. If asked about Charles and Camilla, reply, “Are they here? Has there been an accident?”

10:30 a.m. Summon Andrew and Edward. Inspect hair loss. Close eyes and do “eeny, meeny, miney, mo” to get their hopes up.

11:00 a.m. Watch videotape of Charles’s wedding to Diana. Consider Disney’s request to turn ceremony into stage musical. Messenger DVD of “Shrek” to C. and C.

12:00 p.m. Lunch with corgis. Discuss Blair, Bush, Iraq—what do they think? Show them special surprise: photo mockup of U.N., with all delegates as corgis.

1:00 p.m. Phone Clint Eastwood; congratulate on Oscar. Suggest he next direct “Camilla,” inspirational love story à la “Bridges of Madison County.” Also suggest he play title role.

2:00 p.m. Dress up corgis as participants in low-key royal wedding. Use bits of sirloin to stimulate barking as vows. Videotape and send anonymously to BBC.

3:00 p.m. Nap. Dream of being Virgin Queen, or LaToya Jackson, anything with more dignity. 4:00 p.m. Awake. Ask secretary if C. and C. ceremony has concluded. Ask if Angelina Jolie

was in attendance. 5:00 p.m. Call Charles on cell. Congratulate him, then make connection-breaking-

up noises, so only words he hears are “king,” “never,” and “hee hee hee.” 6:00 p.m. Dinner with corgis. Tell them that C. and C. are now married, just like Britney.

Serve tiny wedding cake made of liver. In honor of ceremony and late Queen Mum, let corgis have bourbon.

7:00 p.m. Watch “Lost.” Wonder if C. and C. will take plane on honeymoon. 8:00 p.m. Call Camilla, to interrupt wedding night. Ask if she has Prince Charles in a can.

Hang up. 9:00 p.m. Read corgis “Cinderella” as bedtime story, but change all characters to corgis, so

happy ending will be believable. Tell them only King Charles will be a spaniel. 10:00 p.m. Put on crown. Take Ambien.

• 2 players• Opposite interests (zero sum)

• communication does not matter• binding agreements do not make sense

• 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

• 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

• 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

• 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

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.

col 1 col 2row 1 $11 $27row 2 $34 $42

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.

col 1 col 2row 1 $11 $27row 2 $34 $42

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.

col 1 col 2row 1 $11 $27row 2 $34 $42

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

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

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.

col 1 col 2row 1 $11 $27row 2 $34 $42

Game Splitting a piece of

cake In two

Statistician Game theorist

In more than two

Team work division Splitting work for

projects

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

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

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

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

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

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

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.

col 1 col 2 worstrow 1 25 67 25row 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.

col 1 col 2 worstrow 1 25 67 25row 2 34 14 14

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.

col 1 col 2row 1 $25 $67row 2 $34 $14worst $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.

col 1 col 2row 1 $25 $67row 2 $34 $14worst $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?

col 1 col 2 worstrow 1 $25 $67 $25row 2 $34 $14 $14worst $34 $67

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

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

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

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

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

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

You must select your strategy randomly!!!

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

Notice that in examples 1 & 2 (which are trivial to solve) we have that

maximin = minimax

col 1 col 2worst (row minimum)

row 1 $11 $27 $11row 2 $34 $42 $34worst (column maximum) $34 $42

maximinMinimax

col 1 col 2worst (row minimum)

row 1 $25 $67 $25row 2 $34 $14 $14worst (column 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

maximin

Minimax

• 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

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?

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

• 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

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.

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

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

• 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

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.

Models(see Word document)