Chapter 15- Game Theory: The Mathematics of Competition · Web viewA higher number is better for Bob and therefore worse for Sally. (Note- In game theory notation, the payoff matrix

Name:___________________________________________________ Date:________ Discrete Math 2: 15.1 Game Theory; Two-person Total Conflict Games (Page 1)

Objective(s): The learner will be able to calculate the maximin and minimax strategies from a two-person total-conflict game. In addition, the student will determine if a saddlepoint arises from the strategies.__________________________________________________________________________________________

Chapter 15- Game Theory: The Mathematics of Competition

Chapter 15 Outline Two-Person Total Conflict Games: Pure Strategies Two-Person Total Conflict Games: Mixed Strategies Partial Conflict Games Larger Games Extensive Form Games: A sequence of moves

Web site for further exploration: www.gametheory.net

DefinitionsGame Theory: use of mathematical tools to study situations (games) involving conflict and cooperation.

Strategies: courses of action a player can choose in a game

Outcomes: consequences of choice made.

Rational Choice: a choice that leads to a preferred outcome

Total Conflict: (a zero-sum or constant-sum game) when one player wins, the other player loses

Introduction- Fundamental Principles of Game Theory

When analyzing any game, we make the following assumptions about both players: Each player makes the best possible move. Each player knows that his or her opponent is also making the best possible move.

A) Two-person Total Conflict Game: Pure StrategiesA player uses a pure strategy if he or she uses the same strategy at each round of the game.

Zero-sum games: Games of “total conflict” – one player’s _________ equals the other player’s ________. If we sum the payoffs at each outcome, the result is always _________.

Two-player simultaneous move games (both zero and variable sum types) can be written in ___________ form (also called __________________ form)

Example: (p470 picture)

http://www.gametheory.net/

Let’s say 2 people (Sally and Bob) are trying to decide on the location of a restaurant near an intersection.Bob wants a higher elevation; Sally wants a lower elevation. They both want an intersection. Sally will choose from the 3 the N-S highways; Bob will choose the 3 E-W highways. The numbers in the chart represent altitude in thousands of feet.

Sally

Bob

A higher number is better for Bob and therefore worse for Sally. (Note- In game theory notation, the payoff matrix in a zero-sum game will always be in terms of the row player) Sally will choose 1, 2, or 3 and Bob will choose A, B, or C. Bob wants to get the highest altitude, no matter what route Sally chooses. Sally wants the lowest altitude, no matter what Bob chooses. Which row do you think Bob should select? _________ Which column do you think Sally should select? _________ Now let’s look at the strategy:

If Bob selects A, Sally will choose _______ , and the value would be _______ . If Bob selects B, Sally will choose _______ , and the value would be _______ .If Bob selects C, Sally will choose _______ , and the value would be _______ .

If Sally selects 1, Bob will choose _______ , and the value would be _______ .If Sally selects 2, Bob will choose _______ , and the value would be _______ .If Sally selects 3, Bob will choose _______ , and the value would be _______ .

Based on the strategies, which row should Bob select? _________Based on the strategies, which column should Sally select? _________

With rows (which are to be maximized), we use the biggest minimum (maximin).With columns (which are to minimized), we use the smallest maximum (minimax).Saddlepoint - when the maximin and minimax strategies produce an equal valueChoosing the row and column through any saddle point gives ______________ strategies for both players.

Examples: Determine if there is a saddlepoint. If so, what is the best strategy?1)

Which row should player 1 pick?Which column should player 2 pick?

Is there a saddlepoint? What is it?

2) Which row should player 1 pick?Which column should player 2 pick?


Name: ________________________________________________________________________________________ Date _____________

Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 2a)

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Player 21 2 3

A 8 2 6Player 1 B 3 4 7

C 5 5 4

Player 21 2 3 4

A 10 17 20 15Player 1 B 15 15 15 25

C 20 20 20 20D 15 15 20 20

For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.

1) (Player 2)(Player 1) Columns

Rows 1 2 3 Which row should player 1 pick?A 40 35 30B 25 30 20 Which column should player 2 pick?C 25 25 30D 20 20 15 Is there a saddlepoint? What is it?


Rows 1 2 3 4 Which row should player 1 pick?A 100 150 300 200B 250 220 250 150 Which column should player 2 pick?



Rows 1 2 3 4 Which row should player 1 pick?A 30 70 40 80B 40 20 10 70 Which column should player 2 pick?C 20 60 70 40D 60 50 30 20 Is there a saddlepoint? What is it?



Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 2b)

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Rows 1 2 3 Which row should player 1 pick?A 3 4 2B 1 4 6 Which column should player 2 pick?C 9 7 8



Rows 1 2 3 4 Which row should player 1 pick?A 1 5 7 3B 6 2 4 5 Which column should player 2 pick?C 3 4 6 7







Name: ________________________________________________________________________________________ Date _____________

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Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 3a)











Discrete Math 2: 15.1 – Game Theory; Two-Person Zero-Sum Games (Page 3b)

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Rows 1 2 3 4 Which row should player 1 pick?A 10 50 70 30B 60 20 40 50 Which column should player 2 pick?C 30 40 60 70







Name:___________________________________________________ Date:________ 6

Discrete Math 2: 15.2 Two person Total Conflict Games, Mixed Strategies (Page 1a)Objective(s): The learner will be able to calculate the optimal probabilities for each strategy for each player in a two-person total-conflict game.__________________________________________________________________________________________B) Two-person total conflict game: Mixed strategies (NO saddlepoint!!)A mixed strategy is a method of playing a game where the rows or columns are played at ___________ so that each is used a given fraction of the time. Pure strategies are the actual strategies that players have available to choose from when playing a matrix game. In a simultaneous move (matrix) game, sometimes players can benefit from randomly choosing one or the other of their pure strategies. A mixed strategy is the decision to play each of the pure strategies with some specific _________________. If pure strategies do not produce a ________________, we proceed as follows: Define variables that represent the __________________ each player will play each available strategy. For each player we find the probabilities that will provide the lowest expected payoff for the other player.

Example:

What is Player’s 1 best strategy?

What is Player’s 2 best strategy?

Is there a saddle point? What does this mean?

1) Let’s look at Player 1 first. What is the best that Player 1 can do, regardless of player 2’s choice?Player 1’s expected payoff is:

So Player 1 chooses Heads _____ of the time and Tails ____ of the time. The expected payoff for Player 1 is2) And Player 2’s expected payoff is:

Player 2 will choose Heads ____ of the time and Tails ______ of the time. The expected payoff for Player 2 is

This was a zero-sum game: one in which the payoff to one player is the negative of the corresponding payoff to the other, so the sum of the payoffs is always zero.Example: When a pitcher and batter face each other in a baseball game, we can consider each pitch as a simultaneous move zero-sum game. In this situation, the pitcher can throw a fastball or a curve ball. The batter

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Player 2

Heads Tails

Player 1

Heads 5 -2

Tails -3 1

will swing after guessing that the pitcher will throw a fastball or a curve ball. The numbers shown represent batting averages. Find the optimal mixed strategy for the pitcher and for the batter.

Pitcher

Which row should the Batter pick?Which column should Pitcher pick?Is there a saddlepoint? What does this mean?

What % should Batter choose Fastball? What % should Batter choose Curveball?

What % should Pitcher choose Fastball? What % should Pitcher choose Curveball?__________________________________________________________________________________________

Example: (Player 2)(Player 1) Columns

Rows 1 2 Which row should player 1 pick?A 47 60 Which column should player 1 pick?B 28 31 Is there a saddlepoint? What does this mean?

Find the mixed strategy for each player: (Player 2) What % should P1 choose row A?

(Player 1) Columns What % should P1 choose row B?Rows 1 2 What % should P2 choose column 1?

A 47 60 q What % should P2 choose column 2?B 28 31 (1 – q)

p (1 – p)

Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.2– Two-Person Zero-Sum Games, Mixed Strategies (Page 2a)

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Fastball Curve

Batter Fastball .300 .200

Curve .100 .500










Rows 1 2 Which row should player 1 pick?A 350 450 Which column should player 2 pick?B 400 150 Is there a saddlepoint? What does this mean?

What % should P1 choose row A?What % should P1 choose row B?What % should P2 choose column 1?What % should P2 choose column 2?

Discrete Math 2: 15.2– Two-Person Zero-Sum Games, Mixed Strategies (Page 2b)

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For the following examples, assume that player 1 picks the row and is trying to maximize the number and player 2 picks the column and is trying to minimize the number.5) (Player 2)

(Player 1) ColumnsRows 1 2 Which row should player 1 pick?

A 6 8 Which column should player 1 pick?B 10 2 Is there a saddlepoint? What does this mean?

What % should P1 choose row A? What % should P1 choose row B?What % should P2 choose column 1? What % should P2 choose col 2?

6) Pitcher

Which row should the Batter pick?Which column should Pitcher pick?Is there a saddlepoint?

What % should Batter choose Fastball? What % should Batter choose Curveball?What % should Pitcher choose Fastball? What % should Pitcher choose Curveball?

7) When it is third down and short yardage to go for a first down in American football, the quarterback can decide to run the ball or pass it. Similarly the other team can prepare a heavier defense against a run or pass. The following matrix displays the probabilities of obtaining a first down.

Defense

Which row should the Offense pick?Which column should Defense pick?Is there a saddlepoint?

What % should Offense choose to Run? What % should Offense choose to Pass?What % should Defense choose Run? What % should Defense choose Pass?Name: ________________________________________________________________________________________ Date _____________

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Fastball Curve

Batter Fastball .400 .200

Curve .100 .500

Run Pass

Offense Run 0.5 0.8

Pass 0.7 0.2

Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 3a)

Find the optimal mixed strategy for each participant:

1) Player 1 is trying to maximize and Player 2 is trying to minimize.

Player 21 2

Player 1 A 5 10 qB 7 6 (1 – q)

p (1 – p)

2) PitcherF C

Batter F .250 .150 qC .200 .400 (1 – q)

p (1 – p)

Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 3b)

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3) It’s a soccer penalty kick! Should the shooter kick to the side or the center and which way should the goalie guess? The number represents the probability of the kick being made. Specify the best strategies for each player.

GoalieSide Center

Shooter Side .4 .9 qCenter .8 .3 (1 – q)

p (1 – p)

4) Two thieves were caught. They must to decide whether to confess or to keep quiet. The numbers listed show the length of sentence each would receive:

Thief 2Quiet Confess

Thief 1 Quiet 0 6 qConfess 6 3 (1 – q)

p (1 – p)

Name: ________________________________________________________________________________________ Date _____________

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Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 4a)

NEW SHORT-CUT: Assume the values are in the grid as: A BC D

p =

(B−D )(C+B )−(A+D) q =

(C−D)(C+B )−(A+D)


Player 21 2

Player 1 A 8 3 qB 4 9 (1 – q)

p (1 – p)

What fraction/percent should:

Player 1 choose A? __________Player 1 choose B? __________Player 2 choose 1? __________Player 2 choose 2? __________

2) PitcherF C

Batter F .400 .200 qC .100 .500 (1 – q)

p (1 – p)


Batter guess Fastball? __________Batter guess Curve? __________Pitcher throw Fastball? __________Pitcher throw Curve? __________

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GoalieSide Center

Shooter Side 50% 80% qCenter 90% 20% (1 – q)

p (1 – p)


Goalie guess Side? __________Goalie guess Center? __________Shooter kick Side? __________Shooter kick Center? __________




p (1 – p)

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Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.2 – Two Person Total-Conflict Games, Mixed Strategies (Page 5a)


Player 21 2

Player 1 A 7 3 qB 2 9 (1 – q)

p (1 – p)


Player 1 choose A? __________Player 1 choose B? __________Player 2 choose 1? __________Player 2 choose 2? __________

2) PitcherF C

Batter F .400 .300 qC .200 .500 (1 – q)

p (1 – p)


Batter guess Fastball? __________Batter guess Curve? __________Pitcher throw Fastball? __________Pitcher throw Curve? __________

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GoalieSide Center

Shooter Side 40% 90% qCenter 80% 30% (1 – q)

p (1 – p)


Goalie guess Side? __________Goalie guess Center? __________Shooter kick Side? __________Shooter kick Center? __________




p (1 – p)

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Name:___________________________________________________ Date:_______Discrete Math 2: 15.3 Two person Partial Conflict Games, Mixed Strategies (Page 1)

Objective(s): The learner will be able to calculate the Dominant Strategy for each player in a two-person partial-conflict game.__________________________________________________________________________________________C) Partial Conflict GamesIn the real world, not every player’s loss is another player’s gain. For example, both may benefit or both may lose. Games of partial conflict are variable-sum games, in which the sum of payoffs to the players at the different outcomes varies.Partial Conflict: a variable-sum game in which both players can benefit by cooperation but may have strong incentives not to cooperateReal-world applications: labor-management disputes, international crises, Arms race

Now, let’s play another game!!

Assuming player I and II move simultaneously, what should they pick?If you are player I, what is your strategy to optimize your payoff? Do you pick 1 or 2? If you are player II, what is your strategy? Is it better to pick A or B?

In this case, both teams have Dominant Strategies. The dominant strategies are Strategy 1 for All-Star(1 is ALWAYS better than 2) and Strategy A for Awesome (A is ALWAYS better than B).

Dominant Strategy- a strategy that is sometimes better and never worse for a player than every other strategy, whatever strategies the other players choose

Example: Does either player have a dominant strategy in the game below?

Player 2 Player 2

1)2)

Player 1 Player 1

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Team Awesome

Strategy A Strategy B

Team All-Star

Strategy 1 (2,2) (4,1)

Strategy 2 (1,4) (3,-3)

A B

X 1,3 1,1

Y 1,2 1,4

Z 1,0 1,1

A B

X 0,1 3,0

Y -5,5 1,3

Z 1,4 6,-3

This leads to a special type of game called the Prisoner’s Dilemma.The game got its name from the following hypothetical situation: imagine two criminals arrested under the

suspicion of having committed a crime together. However, the police does not have sufficient proof in order to have them convicted. The two prisoners are isolated from each other, and the police visit each of them and offer a deal: the one who offers evidence against the other one will be freed. If none of them accepts the offer, they are in fact cooperating against the police, and both of them will get only a small punishment because of lack of proof. They both gain. However, if one of them betrays the other one, by confessing to the police, the defector will gain more, since he is freed; the one who remained silent, on the other hand, will

receive the full punishment, since he did not help the police, and there is sufficient proof. If both betray, both will be punished, but less severely than if they had refused to talk. The dilemma resides in the fact that each prisoner has a choice between only two options, but cannot make a good decision without knowing what the other one will do. Using the matrix above, did either criminal have a dominant strategy?

Let’s play another game!! Does either player have a dominant strategy?

1) This game is an example of a type of game called Chicken. 2) Arms race. Lets say we have 2 countries Chicken: a two-person, variable-sum symmetric game in (Red and Blue) that hate each other. They have 2 which each player has two strategies: to swerve to avoid a choices about how to handle their arms. 1) arm ‘collision’ or not to swerve if the other swerves. in prep for possible war 2) disarm or at least

try to negotiate arms-control agreement Player 2

Red

Swerve

Not Swer

ve

Swerve

(3,3) (2,4)

Not Swer

ve

(4,2) (1,1)

3) The Cuban missile crisis can be modeled by the game of chicken. In the 1960s, the USSR began supplying missiles to Cuba. The United States began a blockade to stop the USSR. As the crisis developed, if each had continued to proceed with their chosen strategy, the consequences could have been disastrous. Fortunately, some last minute negotiations averted such an outcome. The U.S. did not have to back down and the USSR was able to achieve some advantage in other interests.

USSR

back down proceed

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Confess Not Confess

Confess (2,2) (4,1)

Not Confess

(1,4) (3,3)

Continue to Arm

Respect Treaty

Continue to Arm

(2,2) (4,1)

Respect Treaty

(1,4) (3,3)

back down (3,3) (2,4)

proceed (4,2) (0,0)

Name:___________________________________________________ Date:________ Discrete Math 2: 15.3 Two person Partial Conflict Games, Mixed Strategies (Page 2a)

Objective(s): The learner will be able to calculate the Dominant Strategy for each player in a two-person partial-conflict game.__________________________________________________________________________________________Is there a dominant strategy for either player? If so, what is it?

1) Player 2L R

U 10,5 20,9

Player 1 M 20,10 30,5

D 30,10 6,5

2) Prisoner 2Confess Deny

Prisoner 1 Confess -10, -10 -1, -25

Deny -25, -1 -3, -3

3) Player 2X Y

A 5,2 4,2

Player 1 B 3,1 3,2

C 2,1 4,1

D 4,3 5,4

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4) Player 2X Y

A 6,3 4,3Player 1

B 3,1 3,3

C 2,1 4,1

D 4,4 5,4

Discrete Math 2: 15.3 Two person Partial Conflict Games, Mixed Strategies (Page 2b)

5) Player 2L R

Player 1 M 20,10 30,5

D 30,10 5,5

6) Player 2V W X Y Z

A 4,-1 3,0 -3,1 -1,4 -2,0

B -1,1 2,2 2,3 -1,0 2,5Player 1

C 2,1 -1,-1 0,4 4,-1 0,2

D 1,6 -3,0 -1,4 1,1 -1,4

E 0,0 1,4 -3,1 -2,3 -1,-1

7) Player 2L R

Player 1 U 5, 5 2, 1

D 4, 7 3, 6

8) Player 220

L RPlayer 1

U 3, 1 0, 0

D 0, 0 1, 3

9) P2V W X Y Z

A 4,-1 4,2 -3,1 -1,2 -2,0

B -1,1 2,2 2,3 -1,0 2,5P1

C 2,3 -1,-1 0,4 4,-1 0,2

D 1,3 4,4 -1,4 1,1 -1,2

E 0,0 1,4 -3,1 -2,3 -1,-1

Name:___________________________________________________ Date:________ Discrete Math 2: 15.4 Game Theory; Truels (Page 1)

Objective(s): The learner will be able to calculate the Dominant Strategy for each player in a truel game and use trees to determine outcomes of extensive games.__________________________________________________________________________________________D: Extensive Form Games: Game TreesSometimes, the game may not be ‘simultaneous choice’ but instead of sequence of choices (these are called sequential move games and are often modeled using a game tree)

The game tree is referred to as the extensive form of the game.

Tic tac toe, for example, could be modeled in a game tree. (a very very large one)

Example:Let’s say there are 2 monkeys and 1 tree with 5 bananas. There is a Big Monkey and a Little Monkey. One monkey will move first, then the other after.

Let’s use Backwards Induction to solve this.

What are little monkey’s strategies??

What are Big Monkey’s Strategies?

What is the resulting payoff?

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Would this change if we changed the order?

A Truel is like a duel, except there are three players. Each player can either fire, or not fire, his gun at either of the other two players. We assume the goal of each player is, first, to survive and, second, to survive with as few other players as possible. Each player has one bullet and is a perfect shot; no communication (for example, to pick out a common target) leading to a binding agreement with other players is allowed, making the game non-cooperative. We will discuss the answers that simultaneous choices, on the one hand, and sequential choices, on the other, give to what is optimal for the players to do in the truel.

If choices are simultaneous, at the start of play, each player will fire at one of the other two players, killing that player. Why will the players all fire at each other? What should each player do?

This analysis suggests that truels might be more effective than duels in preventing the outbreak of conflict.

The game, and optimal strategies in it, would change if the players (1) were allowed more options, such as to fire in the air and thereby disarm themselves, and (2) did not have to choose simultaneously but, instead, a particular order of play were specified. Thus, if the order of play were A, followed by B and C choosing simultaneously. If a players does not shoot the game ends. What should each player do? Let’s create a game tree.

Key(x, y, z) = (payoff to A, payoff to B, payoff to C).4 = best, 3 = next best, 2 = next worst, 1 = worst.NS = not shoot; S X = shoot X.

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Order may lead to radically different outcomes compared with those based on simultaneous choices.

In 1992, a modified version of this scenario was played out in late-night television programming among the three major TV broadcasting networks of the time, with ABC’s effectively going first with Nightline, its well-established news program, and CBS’s and NBC’s dueling about which host, David Letterman or Jay Leno, to choose for their entertainment shows. What happened?

Name: ________________________________________________________________________________________ Date _____________ Discrete Math 2: 15.4 – Game Theory; Truels (Page 2)

You are in a truel – a duel involving three people instead of two! In this situation, each player (A, B, C) has a gun with one bullet and never misses whatever target he/she selects. Each person’s goal is to survive. No communication is allowed by any of the players, so no collusion is allowed. What would you do as each player in the following scenarios?

1) All players act simultaneously – a shot must be fired and no missing is allowedWhat would you do as: Player A:

Player B: Player C:

What would be the resulting outcome? (Who would survive?)

2) All players act simultaneously – a shot must be fired but missing is allowedWhat would you do as: Player A:

Player B: Player C:


3) All players act simultaneously – one can choose not to fire and missing is allowedWhat would you do as: Player A:

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Player B: Player C:


4) Player A acts first, then B & C act simultaneously – a shot must be fired and no missing is allowedWhat would you do as: Player A:

Player B: Player C:


5) Player A acts first, then B & C act simultaneously – a shot must be fired but missing is allowedWhat would you do as: Player A:

Player B: Player C:


6) Player A acts first, then B & C act simultaneously – one can choose not to fire and missing is allowedWhat would you do as: Player A:

Player B: Player C:


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Documents

Chapter 15- Game Theory: The Mathematics of Competition · Web viewA higher number is better for Bob and therefore worse for Sally. (Note- In game theory notation, the payoff matrix