Game Theory (Part 5)

John Baez

Here are a bunch of puzzles about game theory, to see if you understand the material so far.

Classifying games

For each of the following games, say whether it is:

a) a single-player, 2-player or multi-player game

b) a simultaneous or sequential game

c) a zero-sum or nonzero-sum game

d) a symmetric or nonsymmetric game

e) a cooperative or noncooperative game

If it can be either one, or it is hard to decide, explain why! You will probably not know all these games. So, look them up on Wikipedia if you need to:

1) chess

2) poker

3) baseball

4) rock-paper-scissors

5) rock-paper-scissors-lizard-Spock

6) Prisoner's Dilemma

7) solitaire with cards (also known as "patience")

8) the ultimatum game

9) guess 2/3 of the average

10) Nim

Battle of the Sexes

Suppose that Alice and Bob are going to the movies. Alice wants to see the movie This is 40 while Bob wants to see Zero Dark Thirty. Let's say Alice and Bob each have two strategies:

1. Watch the movie they really want to see. 2. Watch the movie the other one wants to see.

If they both watch the movie they really want, each goes out alone and gets a payoff of 5. If they both watch the movie the other wants to see, they again go out alone and now each gets a payoff of -5, because they're both really bored as well as lonely. Finally, suppose one watches the movie they want while the other kindly watches the movie their partner wants. Then they go out together. The one who gets to see the movie they want gets a payoff of 10, while the one who doesn't gets a payoff of 7. (They may not like the movie, but they get 'points' for being a good partner!)

Call Alice A for short, and call Bob B. Write down this game in normal form.

Prisoner's Dilemma

Now suppose Alice and Bob have been arrested because they're suspected of having conspired to commit a serious crime: an armed robbery of the movie theater!

They are interrogated in separate rooms. The detectives explain to each of them that they are looking at 3 years of jail even if neither of them confess. If one confesses and the other denies having committed the crime, the one who confesses will get only 1 year of jail, while the one who denies it will get 25 years. If they both confess, they will both get 10 years of jail.

Suppose the two strategies available to both of them are:

1. confess to the crime. 2. deny having done it.

Write down this game in normal form, where \( n\) years of jail time counts as a payoff of \( -n.\)

Game theory concepts

Now, for both the Battle of the Sexes and Prisoner's Dilemma games, answer these questions:

a) Is this a zero-sum game?

b) Is this a symmetric game?

c) Does player A have a strictly dominant pure strategy? If so, which one?

d) Does player B have a strictly dominant pure strategy? If so, which one?

e) Does this game have one or more Nash equilibria? If so, what are they?

Zero-sum and symmetric games

For the next two problems, suppose we have a game in normal form described by two \( m \times n\) matrices \( A\) and \( B.\) Remember from Part 2 of our course notes that if player A chooses strategy \( i\) and player B chooses strategy \( j\), \( A_{ij}\) is the payoff to player A and \( B_{ij}\) is the payoff to player B.

a) What conditions on the matrices \( A\) and \( B\) say that this game is a zero-sum game?

b) What conditions on the matrices \( A\) and \( B\) say that the game is symmetric?

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