Chapter 11: Game Theory I: Basic Concepts and Zero-Sum Games

Question 1

Game Theory

Answer 1

Study of decisions where outcome is based on what others do.

Question 2

Example of Game Theory

Answer 2

Chess

Question 3

Why is Chess a good example of Game Theory?

Answer 3

you try to make the most optimal decisions (to either win or avoid disadvantage) in response to your opponent’s move.

Question 4

Common Knowledge of Rationality (CKR):

Answer 4

General assumption that you will consider the best move your opponent will make. If that move gives the opponent an advantages, and disadvantages you, then your opponent can be sure that you will choose whatever move helps you to avoid the disadvantage.

Question 5

In short, this assumption of CKR between two opponents/players will determine what ____________ you make, and how your _____________ will ________________.

Answer 5

choice; opponent; respond

Question 6

Review Row / Col. Supermarket Game of Maximizing Profits (See Notes)

Question 7

The Supermarket Game is an example of the _____________ ______________.

Answer 7

Prisoner’s Dilemma

Question 8

Explain the Prisoner’s Dilemma (See Notes)

Answer 8

• Row and Col. are arrested for drug dealing (trafficking drugs across state lines).
• Row and Col. are placed in separate holding cells.
• The judge says there’s not enough evidence to convict either of the suspects, so she wants one (or both) of the suspects to confess.
• A confession from one = 1yr. sentence for the confessor & 20yrs. for the person that denied the charges.
• A confession from both = 10yr. sentences for both confessors.
• Denial of the charges from both = 2yrs. for both confessors.

Question 9

What should Row do in the Prisoner’s Dilemma? (SEE NOTES)

Answer 9

• Looking at this from the perspective of Row, Col. is either going to confess or not confess.
• Whether or not Col. confesses, it’s best for Row to confess because -10yrs > -20 yrs., and -1yr. > -2yrs.

Question 10

What should Col. do in the Prisoner’s Dilemma? (SEE NOTES)

Answer 10

• Looking at this from the perspective of Col., Row is either going to confess or not confess.
• Whether or not Row confesses, it’s best for Col. to confess because -10yrs > -20yrs., and -1yr. > -2yrs.

Question 11

In ________ situations then, it would be best if ________ and _________ both ________________, leading to a _____yr sentence for each. However this leads to an overall undesirable _________________, considering had both of them ______________ the charges, they each would’ve gotten ______ years.

Answer 11

both; Row; Col.; confessed; 10; outcome; denied; 2

Question 12

Why wouldn’t Row and Col choose the outcome that benefits the both of them? (SEE NOTES)

Answer 12

• For one thing, they’re in separate cells making this decision. They’re making the decisions that are of the most utility to them, individually.
• Principles of rationality lead each Row and Col., to confessing.

Question 13

What exactly is the “dilemma” in the Prisoner’s Dilemma? (SEE NOTES)

Answer 13

The dilemma is that the choice that’s optimal INDIVIDUALLY does not coincide with the choice that is optimal for the GROUP.

• Individually, Col. and Row are better off confessing, but collectively this leads to a worse outcome than they would’ve otherwise had, than if they made the decision to confess/deny charges together.

Question 14

Re-Working the Dilemma: What if Col. and Row did collaborate? (SEE NOTES)

Answer 14

• Let’s pose a situation wherein the judge allowed Row and Col. to meet for an hour in the same cell.
• Row and Col. agree to deny the charges, that way each of them walk away only having to serve 2 years.
• Though they each make this agreement, Row is rational (and knows it’s stupid to not confess). Row is a rational person, not a moral person.
• So, as Row is the first to approach the judge, he confesses.
• Regardless of what happens from here, Row will be fine.
• If Col. is still under the impression that they were both going to deny charges, then Row will only receive 1 yr, whilst Col. serves 20yrs.
• Or perhaps Col. was planning to do the same thing, and cheat–go behind Row’s back and confess. If this were to be the case, now Row is a lot better off than he would’ve been had he followed through with the agreement to deny the charges (-10yrs > -20yrs).
• Whatever Col. does, it’s ALWAYS better for Row to confess!

Question 15

The Prisoner’s Dilemma arises whenever a game is…

Answer 15

Symmetrical.

Question 16

Symmetrical (Prisoner’s Dilemma)

Answer 16

When both players/ opponents face the same strategies and outcomes.

(e.g. There’s a reason Col. would choose to confess, just like Row would: because confessing is more rational than not.)

Question 17

The only way one player could reach the ____________ most ____________ to them, is if one willingly acts _______________. (but no one’s gonna do that, because these games are symmetrical) (SEE NOTES)

Answer 17

outcome; desirable; irrationally

Question 18

if you’re dealing with any ___________________ values in the matrix for a problem in Prisoner’s Dilemma, the values are always _________________. REMEMBER THAT! However, remember that if you’re using a mixed strategy, these values are _________________ utility values, because then you’re using ________________ _________________ ___________________.

Answer 18

numerical; ORDINAL; interval; maximizing expected utility

Question 19

Zero-sum Game v. Nonzero-sum Game

Answer 19

Zero-sum Game: (fixed amount of what) Player wins as much as opponent loses.

Nonzero-sum Game: Player does not win as much as the opponent loses. (Player doesn’t win an amount equal to what the opponent loses).

Question 20

Noncooperative Game v. Cooperative Game

Answer 20

Noncooperative Game:
- Doesn’t allow players to form a binding agreement, so that the players CAN cooperate.

• This way, players can talk to each other about the strategies / acts they intend to choose, without recourse from a player if another player deviate from the strategies they said they would use.

Cooperative Game:
- Forces players to keep to their agreements/strategies they said they would use in a binding agreement.

Question 21

Simultaneous Move Game v. Sequential Move Game

Answer 21

Simultaneous Move Game:
- Players do not know what choice either player is going to make until it’s made.

Sequential Move Game:
- Players have have SOME or FULL information about the other players choices from previous rounds.

Question 22

What is “Perfect Information”?

Answer 22

When a player has (exactly) FULL information about another players choices (strategies/acts) from previous rounds.

(i.e. Chess)

Question 23

Symmetric Games v. Nonsymmetric Games (SEE NOTES for Matrices)

Answer 23

Symmetric Games:
- When both players face the same strategies and outcomes.

Nonsymmetric Games:
- When both players do not face the same strategies and outcomes.

Question 24

Two-Player Games v. n-Player Games

Answer 24

Two Player Games:
- Games played by two people.
- (multiple people may represent one person [i.e. multinational companies]. It doesn’t matter how many people there are, just how many are playing the game).

n-Player Games:
- Game played by “n” number of people.
- (Harder to illustrate graphically because it can’t be represented cleanly on a decision matrices).

Question 25

Pure Strategy v. Mixed Strategy

Answer 25

Pure Strategy:
- Specifies all choices that player can make (including those relative to another player’s decisions).

Mixed Strategy:
- Pure strategies with probabilities between 1 and 0 (where p>0).

Question 26

How might one apply the mixed strategy to the Prisoner’s Dilemma

Answer 26

Well, let’s remember the pure strategy specifies all choices the player can make: for us that includes confessing and not confessing.

The mixed strategy tells us to apply probabilities from 1-0 to our pure strategies.

So we could apply a probability “p” to confessing, and “1-p” to not confessing (denying the charges).

Question 27

Infinitely Repeated Games, Non-Iterated Game, Iterated Games

Answer 27

Infinitely Repeated Games:
- Games repeated/played an infinite number of times.

Non-Iterated Games:
- Games only played once (no matter how many strategies were available).

Iterated Games:
- Games played multiple times.

Question 28

Tit-for-Tat Strategy

Answer 28

When a player uses the exact strategy their opponent used in the previous round.

Question 29

Many games can be solved by applying the _________________ ________________ in a clever way. Of course, this does require that we adopt a few technical _______________ first.

Answer 29

Dominance Principle; assumptions

Question 30

1st Assumption

Answer 30

All players are rational.

(all players are going to make choices that help them accomplish a certain objective they consider important.)

Question 31

2nd Assumption

Answer 31

All players know all other players are rational.

Question 32

3rd Assumption

Answer 32

The Dominance Principle is a valid principle of rationality.

Question 33

nth Order of Common Knowledge of Rationality

(nth order CKR)

Answer 33

each player knows, that the other player knows, that the first player knows, that the other players know, that each player knows, that the player in question is rational.

Question 34

What does it mean to “solve” a game?

Answer 34

Find what strategies the player will use in the game in question.

Question 35

Review Dominance Principle Problem from notes (SEE NOTES)

Question 36

Unfortunately, the ________________ ________________ won’t work in all games, especially those wherein the player(s)’ don’t have a _________________ strategy, which could lead to _________________, potentially ____________________ outcomes. This can be seen in the _______________ Game.

Answer 36

dominance principle; dominant; undesirable; unacceptable; Centipede

Question 37

Centipede Game (SEE NOTES)

Answer 37

Shows us that the dominance principle doesn’t work in games that don’t have a dominant strategy.

Rules:
- There’s a “dollar pot.”
- Each player has the opportunity to take what’s in the pot, or allow more money to be put in the pot.
- The game lasts 4 rounds.

Scenario:
- Player A and B are opponents.
- There is $1 in the pot
- Player A may either take the $1, or allow another $1 to be placed in the pot.
- Player A allows another dollar to be placed in the pot.
- Player B is given the alternatives of taking the $2 in the pot or allowing another $2 to be placed in the pot.
- Player B allows another $2 to be placed in the pot.
- Player A may take the bulk of the $4 ($3) and leave $1 for Player B, or he could allow another $2 to be placed in the pot.
- We’re now on the 4th and Final Round. Player B can take the bulk of the $6 ($4) and leave $2 for Player A, or evenly divide the profits so each player gets $3.

By REASONING BACKWARDS, we realize due to the Dominance Principle (and that the players are rational)…
- Player B will take the $4 bulk over the even divide of $3, bc $4 > $3.
- Player A knows this, and will take the $3 bulk from the 3rd round, to 1) kill the 4th round, and 2) eliminate the possibility of receiving $2 in the 4th round.
- Player B knows this, and decides to take the $2 in the second round.
- Player A knows this, and takes the first $1 in the 1st round to kill any chance of Player B getting $2 and so Player A doesn’t walk away with nothing.

In all Player A’s and B’s thinking, which the Dominance Principle encouraged them to do, A and B ultimately ended up with an outcome that is overall less desirable. Had they let the game continue to the 4th round, they would’ve each received $3 even (if they chose to). This necessarily means, the Dominance Principle, a principle of rationality, says the dominant act is to receive $1 over $3.

Question 38

The most researched types of games are _________-_______ __________-___________ games.

Answer 38

Two-Person Zero-Sum

Question 39

Two-Person Zero-Sum Game

Answer 39

Game played by two people, and where one player receives as much utility as their opponent loses.

Question 40

Two-Person Zero-Sum Games are actually the _______________ to represent on a decision _______________. You really only have to represent the _______________ of _________, because we know whatever the ______________ of _______ is, that of _______ will just be the _________________. Naturally (and Remember This), Row seeks to ____________ the numbers in the matrix, whereas Column seeks to _________________ the numbers in the table. (SEE NOTES)

Answer 40

easiest; matrix; outcome; Row; outcome; Row; Col.; opposite; maximize; minimize

Question 41

Essentially, if you know the _____________ of ______, you know the _________________ of _______ (which, again, is just the ________________). (SEE NOTES)

Answer 41

outcome; Row; outcome; Col.; opposite

Question 42

Equilibrium

Answer 42

A pair of strategies are in Equilibrium iff. when reaching a certain outcome, neither player can reach a better outcome, BY UNILATERALLY SWITCHING STRATEGIES.

(meaning, one player changes the strategy they’re using–they can’t do that).

Question 43

Minimax Condition

Answer 43

A pair of strategies are in Equilibrium if the outcome equals the minimal value of the Row, and the maximal value of the Column.

Question 44

Minimax Condition is a way of determining what _______________ player(s) uses when making a choice for the game in question: the _______________ value on a Row does the ___________ _______ for Col., and the ________________ value on a Column, does the ___________ _______ for a Row. (SEE NOTES)

Answer 44

strategies; minimal; least bad; maximal; least bad

Question 45

Nice as the Minimax Condition Rule is, it _________ work for _____ games that are _______-___________ ___________-_______. This is because, not every game has a _________ equilibrium _______________- a value on the decision matrix that is both the ______________ value of the Row, and the _______________ of the Column.

Answer 45

doesn’t; ALL; Two-Person Zero-Sum; pure; strategy; minimal; maximal

Question 46

How do we solve this predicament regarding the Minimax Condition Rule?

(Say Answer then GO OVER THE PROBLEM IN IT’S ENTIRETY IN THE NOTEBOOK!!!)

Answer 46

• RANDOMIZATION
  You flip a coin and apply equiprobable probabilities to each player’s mixed strategies, so that you can begin to take the expected utility of each player’s mixed strategy.

Question 47

The Equilibrium utility of a ______-___________, ______________, _________-__________ game must be ____.

Answer 47

two-person; symmetric; zero-sum; 0