This document discusses game theory concepts including matrix games, game trees, and examples like the prisoner's dilemma. Matrix games involve two players secretly choosing rows and columns from a payoff matrix. Game trees model sequential games and examine all possible moves and outcomes. Examples analyzed include the prisoner's dilemma, coordination games, and coin toss games represented as both matrices and trees.

1. Game Theory General Games
Alyaa Muhi

2. Outline
 7.1. Utility
 7.2. Matrix Games
 Coordination Game
 Dating Dilemma
 Volunteering Dilemma
 Stag Hunt
 Competing Pubs
 7.3. Game Trees
 Coin Toss
 Coin Poker
 7.4. Trees vs. Matrices

3. Matrix Game
 A matrix game is a game played between Rose and Colin
using a fixed matrix 𝐴, known to both players, where each
entry of 𝐴 consists of an ordered pair (𝑎 , 𝑏) where 𝑎
indicates the payoff to Rose and 𝑏 the payoff to Colin.
 Rose secretly chooses a row.
 Colin secretly chooses a column .
 If this row and column selects the entry (𝑎 , 𝑏) from the
matrix 𝐴, then Rose gets a payoff of 𝑎 and Colin gets a
payoff of 𝑏.

4. Matrix Game
A B C
A (2, 2) (0, 0) (-2, -1)
B (-5, 1) (3, 4) (3, -1)
Player 1
Player 2Strategy set
for Player 1
Strategy set
for Player 2
Payoff to
Player 1
Payoff to
Player 2

5. Prisoner’s Dilemma
 Rose and Colin have been caught robbing a bank, but the
police don’t have all the necessary evidence to charge them
with the maximum penalty.
 The police isolate the players and offer each the option to
give evidence to convict the other and, in return, receive less
jail time.
 So each player can either cooperate with the other player (C)
and stay silent or may defect (D) by turning over evidence.

6. Prisoner’s Dilemma
Rose
Colin
CONFESS
CONFESS
-1 -1

7. Prisoner’s Dilemma
Rose
Colin
CONFESS
Denny
-10 0

8. Prisoner’s Dilemma
Rose
Colin
Denny
Denny
-5 -5

9. Prisoner’s Dilemma
Rose
Colin
Denny
0 -10
CONFESS

10. Prisoner’s Dilemma
Rose
Colin
Denny
0 -10
CONFESS
CONFESS
Denny
-5 -5
-10 0-1 -1

11. Utility
 Suppose the following matrix gives Rose’s utility for each of the four outcomes:
 If she buys insurance, her expected utility is:
 On the other hand, if Rose doesn’t buy insurance, her expected utility is
(9/100)-(0)+(1/100)(-300,000)= $ -3,000.
(9/100)-(-1,000)+(1/100)(-51,000)= $ -1,500

12. Utility
 Suppose the following matrix gives Rose’s utility for each of the four outcomes:
 I If Rose chooses the sure money, then she gets.
(1/2)(1,000,000)+(1/2)(1,000,000)= $1,000,000
 On the other hand, if Rose doesn’t buy insurance, her expected utility is
(1/2)(2,200,000)+(1/2)(0)= $1,100,000

13. Coordination Game
X Y
X 1, 1 0, 0
Y 0, 0 1, 1
Colin Rose and Colin are test subjects in a psychology
experiment. They have been separated, and each player
gets to guess either 𝑋 or 𝑌. Both players get $1 if their
guesses match and nothing if they do not.
 In a game such as this one, communication between the
players would result in an advantageous outcome. If the
players knew the game and were permitted to
communicate prior to play, it would be easy for them to
agree to make the same choice.
X Y
X 1, 1 0, 0
Y 0, 0 1, 1
Rose

14. Dating Dilemma
B F
B 2, 1 0, 0
F 0, 0 1, 2
Rose
Colin Each player must individually decide to go to the Ball game (B) or to
the Film (F).
 The players prefer to spend the evening together, so payoffs where
the players are in separate places are worst possible for both
players. The tricky part of this dilemma is that Rose would prefer to
end up with Colin at the Ball game, whereas Colin would rather be
with Rose at the Film
 Suppose that Rose committed to going to the ball game and Colin
knew of this decision. Then his best move is to attend the ball
game, too, giving Rose her favorite outcome.

15. Volunteering Dilemma
S V
S -10,-10 -2,-2
V -2,-2 -1,-1
Rose
Colin Each has the option of either volunteering (V) to do the
dishes or staying silent (S).
 If neither player volunteers to do the dishes, the payoff is
quite bad for
 Each player would most like to stay silent and have the
other volunteer.
 The situation when two people are staring at one another,
each hoping the other will volunteer to do something that
both want done but neither wants to do.

16. Stag Hunt
S R
S 3,3 0, 2
R 2,0 1, 1
Rose
Colin
 Rose and Colin are headed off to the woods on a
hunting trip.
 Each player has two strategies work together and hunt
for a stag (S) or go for a rabbit alone (R).
 Obviously, both players do best here if they cooperate
and hunt the Stag. Really the only sticky point is that if
one player suspects the other may go for a rabbit, then
that player has incentive to choose R, too.
Communication is likely to help here, as long as the
players trust each other enough to cooperate.

17. Movement Diagram
 For each column, draw an arrow from the outcome Rose likes least
to the one she likes best (in case of a tie, use a double headed
arrow).
 For each row, draw an arrow from the outcome Colin likes least to
the one he likes best (in case of a tie, use a double headed arrow).
Figure . Movement diagrams for our dilemmas

18. Game Trees
 A game tree is a type of recursive search function that examines all possible moves
of a strategy game, and their results, in an attempt to ascertain the optimal move.
They are very useful for Artificial Intelligence in scenarios that do not require real-
time decision making and have a relatively low number of possible choices per
play. The most commonly-cited example is chess, but they are applicable to many
situations.

19. Game Trees Example Player 1
 Strategy set for Player
1: {L, R}
Player 2 Player 2
L
L
R
RR L
3, -3 0, 0 -2, 2 1, -1
 Strategy for Player 2: __, __
what to do when
P1 plays L
what to do when P1
plays R
 Strategy set for Player 2: {LL,
LR, RL, RR}
Payoff to
Player 2
Payoff to
Player 1

20. Coin Toss
Rose
Colin
Figure: A game tree for Coin Toss

21. Coin Poker
 Rose and Colin each put one chip in the pot as ante and each player
tosses a coin .
 Rose sees the result of her toss, but not Colin’s, and vice versa. It is
then Rose’s turn to play, and she may either fold, ending the game
and giving Colin the pot, or bet and place 2 more chips in the pot.
 If Rose bets, then it is Colin’s turn to play and he may either
fold, giving Rose the pot, or he may call and place 2 chips in
the pot. In this latter case, both coin tosses are revealed.
 If both players have the same coin toss, the pot is split
between them. Otherwise, the player who tossed heads wins
the entire pot.

22. Coin Poker
Figure . A game tree for Coin Poker

23. Coin Poker
Figure . Colin’s strategy H

24. Trees to Matrices

25. Matrices to Trees