Chapter 12: Game Theory II: Nonzero-sum and Cooperative Games

Question 1

Nonzero-sum Games

Answer 1

Games where the fixed amount of utility one player wins is not the same as the amount of utility the opponent loses.

Question 2

Example of a Nonzero-sum Game

Answer 2

Prisoner’s Dilemma.

Question 3

Why is Prisoner’s Dilemma a Nonzero-sum Game?

Answer 3

• The outcome of one player’s strategy is not the opposite for the other player.
• In a zero-sum game the outcome of one player should be the opposite (negative value) for the other player such that the equilibrium utility = 0.
• In the Prisoner’s Dilemma, if one prisoner receives 1 yr, the other player doesn’t receive -1 yr., they receive 20 years.
• Perhaps the most important point as to why the Prisoner’s Dilemma is a nonzero-sum game, is because each outcome for the players are going to be NEGATIVE.
• The outcomes of Row and Col. will never be zero because they are all negative (-10, -10 / -2, -2 / -1, -20 / -20, -1)

Question 4

Find the Equilibrium in Prisoner’s Dilemma (SEE NOTES)

Question 5

Explain the Stag Hunt Game:

Answer 5

• There are 2 pre-historic hunters, Row and Col.
• Each have the option of hunting stag or hare.
• There are more bountiful gains that come with hunting stag. However, hunting stag requires that both hunters hunt the stag.
• If both hunters agree to hunt the stag, they’ll each receive 25 kilos of meat.
• Stag hunting CANNOT be done alone. Hunting stag alone will result in 0 kilos of meat.
• The hunters may also choose to hunt hares individually. This is not a task that needs to be done collectively.
• If the hunters agree to hunt hares by themselves, they’ll each receive 5 kilos of meat.

Question 6

Find the Equilibrium Strategy in the Stag Hunt, then flip card over to see if your answer was correct (SEE NOTES)

Answer 6

There are actually TWO equilibrium strategies: (R1, C1) stag-stag, and (R2, C2) hare-hare.

Question 7

In both (____,____) and (____,____), the strategies by both _____________ are ____________ against the strategies of the other __________, where ____________ means means _____-_________-______-________-_____.

Answer 7

(R1, C1); (R2, C2); players; optimal; player; optimal; at-least-as-good-as

Question 8

Is the Stag Hunt a Nonzero-sum game? Answer yes or no, then explain why.

Answer 8

Yes, the Stag Hunt is a Nonzero-sum Game.
- The values of Row’s strategies are not the opposite (negative) values for Col.’s values.

(5,0)
(0,5)
(5,5)
(25,25)

Question 9

(R1, C1) and (R2, C2) are ____________ strategy _________________. Calculate and find the mixed strategy equilibrium, where the probability assigned to Row and Col.’s strategies is 1/5.

Answer 9

pure; equilibria

To find the equilibrium between the players mixed strategies, we first have to find the probability assignment that would leave Row and Col. indifferent between the expected utility of his mixed strategies, we have to first find the expected utility of his mixed strategies.
- [C1 1/5, C2 1/5]
- p = C1 & p = C2
- 1-p (unity compliment)
- EU(C1) = (25p) +(0 * (1-p))
= (25p)
- EU(C2) = (5p) + (5*(1-p))
= (5p) + (5 - 5p)
= 5 -5p + 5p
= 5
- EU(C1) = EU(C2)
- 25p = 5
- p = 5/25 ~ 1/5
- Since the game is symmetric, such that row and col. face the same strategies and outcomes, 1/5 is also the probability of Row’s mixed strategy equilibrium (playing R1 1/5, and R2 1/5).

Question 10

What would a Risk Averse decision maker (hunter) choose during the Stag Hunt.

Answer 10

The risk averse hunter would choose only to hunt for Hare. By hunting for hare, the decision maker knows he will guaranteed leave with something.

• Hunting stag poses a moderate risk. Let’s say one hunter is adamant (strategies remain fixed) on hunting stag. If you decide to hunt for hare instead of the stag (thereby defecting since it’s a noncooperative game), you will get 5 kilos, and now the other hunter has run the risk of receiving nothing (0 kilos).

Question 11

Pareto Efficiency

Answer 11

A state is Pareto efficient iff. the utility level of one player can’t be increased unless the utility level of the other player is decreased.

Question 12

Explain why (R2, C2) the Hare-Hare pure strategy in the Stag Hunt is not Pareto Efficient.

Answer 12

(R2,C2) is NOT pareto efficient because the players could unilaterally switch to different strategies, specifically Stag-Stag (R1, C1), where the utility level of BOTH players are increased (one of the players is increased without the other player’s utility level being decreased).

Question 13

Explain why (R1, C1) the Stag-Stag pure strategy in the Stag Hunt is Pareto Efficient.

Answer 13

(R1,C1) is Pareto Efficient because the players can switch to different strategies wherein one player would have an increased level of utility (5), and the other player would face a decreased level of utility (0).

Question 14

Explain Battle of the Sexes Game (SEE NOTES)

Answer 14

• Row (Wife) and Col. (Husband) are newly weds.
• Now that they’re married, they want to maximize on spending time together. They’re deciding between attending the Opera, or going to the Football Game.
• Row would rather attend the Opera than go to the Football Game.
• Col. would rather go to the Football Game than attend the Opera, BUT is willing to go to the Opera in case Row tags along.
• In this game, Row and Col. spending time together is more desirable/ of more utility than not spending time together.

Question 15

Based on Decision Matrices, find the equilibrium for the Battle of the Sexes (SEE NOTES).

Answer 15

There are actually 2 Equilibria!

Question 16

Now that we know there are 2 pure strategy equilibria for Battle of the Sexes, calculate and find the Mixed Strategy Equilibrium where R1 2/3, C1 1/3.

Question 17

Why don’t we assume the from the start that Row and Col, are going to reason the same during Battle of the Sexes? Explain why.

Answer 17

Okay, well, let’s reason as Row for a minute.
- if we look at the matrix, I would rather choose R1 because getting utility 2 is more desirable than getting utility 1 (by unilaterally switching strategies ending up w/ outcome of C2 –> going to football game).

• Now let’s reason that Col, my husband, reasons the same. If this were the case, he would opt to choose C2 (Football) because 2 utils is better than 1 (which he would’ve received if he had chosen C1 –> the opera).
• If we converge on this outcome in the matrix, where Row chooses R1 (going to the Opera) and Col chooses C2 (the football game), this outcome has an expected payoff of (0,0), meaning they wouldn’t be spending any time together which is overall undesirable.

Question 18

Is Battle of the Sexes a Nonzero-sum Game?

Answer 18

Yes, because the value/utility that one player receives in an outcome isn’t the opposite (negative) value that they’re opponent loses.

Question 19

Explain the Chicken Game (SEE NOTES):

Answer 19

• Row and Col. are driving straight towards each other at really high speeds.
• Neither of them have plans to slow down. They are going to crash into each other and die.
• Utility of Death = -100
• However, if one player cowardly serves out of the way, they are called a “chicken.”
• Utility of being called a chicken = -1
• Utility of the driver that didn’t swerve out of the way (didn’t fold) is 1.
• Utility of both drivers cowering out, and swerving out of each other’s way is 0.

Question 20

Based on Decision Matrix, what is the equilibrium in the Chicken Game? (SEE NOTES)

Answer 20

There are actually 2 equilibria.

Question 21

In this game, each driver faces a much more _______________ outcome, by choosing ________________ strategies, such that they don’t follow through with ____________ into each other: this is an _______-________________ game.

Answer 21

desirable; different; driving; anti-coordination

Question 22

Coordination Game:

Answer 22

Game where players reach best outcome possible by cooperating.
- E.g. Prisoner’s Dilemma, Stag Hunt, Battle of Sexes

Question 23

Anti-Coordination Game:

Answer 23

Game where players reach best outcome possible by NOT cooperating.
- E.g. Chicken Game

Question 24

All games we’ve looked at so far are ______________________ games. Let’s now turn our attention to a _________________ game: The _________________ ______________.

Answer 24

noncooperative; cooperative; Bargaining Problem

Question 25

Explain Rules of Bargaining Problem (SEE NOTES)

Answer 25

• Row and Col. are offered to split $100.
• Each of them are allowed to write down their demands and place their demands in a sealed envelope.
• If either player ask for an amount that would exceed $100 limit, then they will receive nothing.
• The players are allowed to make binding agreements.

Question 26

Explain Less Complicated Version of Bargaining Problem (SEE NOTES FOR DECISION MATRIX):

Answer 26

• Row and Col are offered to split $100.
• Either Row has to take $80 and Col. has to take $20, or Row has to take $20 and Col. has to take $80.
• Otherwise neither of them will be paid.

Question 27

Find Equilibrium of Less Complicated Version of Bargaining Problem (SEE NOTES):

Question 28

Realizing the less complicated version of the problem has ____ pure strategy _______________, the fact that it has a ___________ mixed strategy, the respective outcomes, THE FACT that the players most ____________ outcomes result from working _____________, this ________________ game seems an awful lot like the ____________________ game ______________ of the ________________. From this we realize maybe the difference between _________________ and _____________________ games isn’t all that important, and that _______________ games can essentially be reduced to ___________________ games. You can’t know what makes a binding agreement in a ___________________ game if you don’t know what constitutes a ____________________ game first.

Answer 28

2; equilibria; mixed; desirable; together; cooperative; noncooperative; Battle of the Sexes; cooperative; noncooperative; cooperative; noncooperative; cooperative; noncooperative

Question 29

Explain General Version of Bargaining Problem (SEE NOTES):

Answer 29

• Row and Col. are offered to split $100.
• Let’s suppose I’m Row, and Col. submits a demand for $99.
• I can’t request for more than $1 because that exceeds the $100, and I wouldn’t receive anything.
• So the only possible (and plausible) response is to demand $1.

Question 30

What’s the problem in the Bargaining Game here? Why might we have trouble SOLVING the game?

Answer 30

There’s several different splits of money between both players
(60/40, 70/30, 80/20, 90/10, 99/1, 75/25).

Question 31

Given the potential for several different splits of money between both Player A (Row) and Player B (Col), how would one solve the Bargaining problem (what strategy will the players employ for the game in question)?

(SEE NOTES FOR EXAMPLES)

Answer 31

The basic principle Nash established for solving the Bargaining Problem is (Based on Four axioms) that Player A receives $X, and therefore gets u($X). Player B receives $100-X, and therefore gets v($100 - X).
- The solution to solve bargaining problem is for both players to find the X that maximizes u($X) * v($100-X).

Question 32

Explain the Ultimatum Game:

Answer 32

Players 1 and 2 are offered to split $10.
Player 1 offers to pay Player 2 $X from the $10 sum.
Player 1 will keep $10-X (ten dollars minus the x amount paid to Player 2).
If Player 2 accepts, Player 1’s plan goes accordingly.
If Player 2 rejects, then both players receive nothing and the money is set ablaze.

If the player’s desired outcome is some amount of money, then Player 2 should accept no matter what the offer is:
$0.01 > $0

Question 33

Two Types of Iterated Games:

Answer 33

Finitely Iterated - Games
Infinitely Iterated - Games

Question 34

Finitely Iterated Games

Answer 34

Games iterated/repeated a finite number of times, SUCH THAT WHEN THE PLAYERS ENTER THE FINAL ROUND, THEY KNOW THEY ARE ON THE FINAL ROUND.

Question 35

Explain what a Finitely Iterated Version of Prisoner’s Dilemma might look like.

Answer 35

• First, when both Row and Col. make it to the final round, they know they’re in the final round.
• Up to this point, Row and Col. have cooperated (denied the charges) in all initial rounds.
• Now that both players have reached the final round, either player should defect, and confessing, playing the dominant strategy.
• Whichever player does this will be better off regardless of what the other player decides to do.
• However, the other player knows this, and using BACKWARDS INDUCTION will therefore defect in the penultimate round, confessing.
• The original player knows this and using backwards induction, will confess in the third to last round and so on and so forth until someone confesses in the first round.
• we soon realize, playing the Prisoner’s Dilemma as a one shot game, and as a finitely iterated game is the SAME THING.

Question 36

Infinitely Iterated Games

Answer 36

Games iterated/repeated an infinite number of times. HERE TIME IS IRRELEVANT, THE PLAYERS HAVE NO KNOWLEDGE OF THE LAST ROUND.

Question 37

Explain what an Infinitely Iterated Version of Prisoner’s Dilemma might look like.

Answer 37

• Both Row and Col. have no knowledge of a final round, they under the impression that they’re just going into “the next round.”
• Because the game last indefinitely, Row and Col. would be better off cooperating (denying charges) every round.
• Defect from either player may result in the other player “punishing” the defector by not cooperating in the next round(s).
• Here, it’s best each player employ the “TIT-FOR-TAT” strategy, such that one player uses the exact same strategy their opponent used in the previous round.
  - if my opponent cooperates, I cooperate.
  - if my opponent defects, I defect.
  - if my opponent returns to cooperation in next round, I return to cooperation.

Question 38

Grim-Trigger Strategy

Answer 38

• Type of Tit-for-Tat strategy.
• In all initial and subsequent rounds, I cooperate, as long as my opponent cooperates.
• The moment my opponent defects, I defect in ALL FUTURE rounds.
• Essentially, defect by opponent is not forgiven…ever.

Question 39

Evolutionary Stable Strategy (ESS)

Answer 39

Strategy that is uninvadable against candidate “invading” group, that plays an altogether different strategy.

Question 40

Every _______________ _______________ _______________ is in ______________ _______________, but not every ________________ _________________ has an ________________ ________________ ______________.

Answer 40

Evolutionary Stable Strategy; Nash Equilibrium; Nash Equilibrium; Evolutionary Stable Strategy

Question 41

Evolutionary Stable Equilibrium

Answer 41

A player’s (biological subgroup’s) mixed strategies are in Evolutionary Stable Equilibrium iff. the expected utility of their mixed subgroups are higher than the expected utility of their opponents mixed strategy.

(player won’t unilaterally switch strategies, regardless of what strategy their opponent decides on.)

Question 42

Requirements for Evolutionary Stable Strategy (2)

Answer 42

Preface: EU(x,y) symbolizes expected utility of playing strategy x, against an opponent playing strategy y.

A strategy is in ESS iff.
1) EU(x,x) ≥ EU (y,x) for all y.
2) EU(x,x) > EU (y,x) OR EU(x,y) > EU(y,y) for all y.

Question 43

Review Hawk-Dove Game and do all required mathematics