So... What’s Game Theory?
Game theory refers to a branch of applied math that deals with the
strategic interactions between various ‘agents’, or players.
Agents choose strategies based on information about the
strategies of other players in an attempt to maximize the return
of their actions.
Provides a formal model for economic social situations that
involve the interactions of two or more parties in negotiations.
Slide 1 of 13
Let’s Lay Down Some Framework…
Simultaneous vs. Sequential: Simultaneous games are those in
which players act without knowledge of previous actions or
outcomes. Sequential games refer to those in which players
have some knowledge of previous actions
Perfect Information: all players know the moves previously
made by all other players (examples include chess, go, and
mancala)
Slide 2 of 13
Games Can Get Funky
Symmetric Games: Payoffs depend only on the strategies
employed, not on the players employing them (examples:
prisoner’s dilema, chicken)
Asymmetric: non-identical strategy sets for each player
(example: dictator game)
Zero Sum Games: applicable to real-life situations; players can
neither increase nor decrease available resources, but rather
benefit at the expense of other players (examples: poker,
chess)
Non-zero games: combinations of strategies have a net sum
that doesn’t = zero; one player’s gain does not necessarily
correspond with another’s loss (example: prisoner’s dilemma)
Slide 3 of 13
Nash Equilibrium
Solution concept for a game involving two or more players
No player benefits from changing his or her strategy while
other strategies remain unchanges
Player A is making the best decision A can, taking into account
Player B's decision, and B is making the best decision B can,
taking into account A's decision.
Slide 4 of 13
How are Games Represented?
Normal Form (Matrix)
Used for games with only one
decision by each player
N player game = N-Dimensional
matrix, with the payoffs for each
player in the cell
To the right, general matrix for a 2player, 2-strategy game
(Player 1)
(Player 1)
Strategy
Strategy
A
B
P(A,A)
P(B,A)
P(A,B)
P(B,B)
(Player 2)
Strategy A
(Player 2)
Strategy B
Slide 5 of 13
How are Games Represented?
Extensive Form (Tree)
Used when order of decision is important
or when each player has more than one
move
Used for many commonly known games




Chess
Checkers
Go
Tic-Tac-Toe
Example 2-player, 2-branching tree on
right
Slide 6 of 13
Prisoner’s Dilemma - Intro
Two criminals can turn evidence on the other or
remain silent
If they both cooperate (remain silent), they
each only get 1 year
(Player 1)
(Player 1)
Cooperate
Defect
1,1
0,5
0,5
3,3
If one cooperates and one defects (turns
evidence), the one that defected gets 0 years
and the one that cooperated gets 5 years
If they both defect, they each get 3 years
What is the best strategy?
(Player 2)
Cooperate
(Player 2)
Defect
Slide 7 of 13
Prisoner’s Dilemma - Analysis
If other player cooperates, defecting gives less time in jail
(0 vs. 1)
(Player 1)
(Player 1)
If other player defects, defecting gives less time in jail (3
vs. 5)
Cooperate
Defect
1,1
0,5
0,5
3,3
Defecting is a dominant strategy (best no matter what)
What does this mean?
For any single game, defection is the best option
If the games are iterative, but independent (i.e. it is impossible
for your strategy in one game affect your opponent’s strategy
in another game), defection is the best option
If games are iterative, but not independent?

(Player 2)
Cooperate
Becomes a tree-based, multi-move game to which this analysis no
longer applies
(Player 2)
Defect
Slide 8 of 13
Pirate Game:
A Sample Complex Analysis
A group of pirates splits up loot in the following manner
Pirates are ranked #1 through #N by seniority
Pirate 1 makes a proposal for splitting gold, which is voted on (“Yes” or “No”) by
all pirates
If at least half vote yes, proposal passes
Otherwise, Pirate 1 is thrown overboard, everyone moves up a rank, and the
process repeats
Assuming pirates are rational and wish to maximize, in order of importance,
survival, wealth, and rank, how would 5 pirates split up 100 gold pieces?
Slide 9 of 13
Pirate Game:
A Sample Complex Analysis
Of the proposals by pirate 1 which pass, which maximizes his
utility?
For at least 2 other pirates, utility from “Yes” must be more
than utility from “No”
Utility from “Yes” can be set by pirate 1
What is utility for pirates 2-5 for saying “No”?

Utility for those pirates from Pirate 2’s proposal
Slide 10 of 13
Pirate Game:
A Sample Complex Analysis
Work backwards
Pirate 4 – Needs 0 extra votes

Distribution is {0,0,0,100,0}
Slide 11 of 13
Pirate Game:
A Sample Complex Analysis
Work backwards
Pirate 4 – Needs 0 extra votes

Distribution is {0,0,0,100,0}
Pirate 3 – Needs 1 extra vote


Can buy vote from 5 for 1 gold
Distribution is {0,0,99,0,1}
Slide 12 of 13
Pirate Game:
A Sample Complex Analysis
Work backwards
Pirate 4 – Needs 0 extra votes

Distribution is {0,0,0,100,0}
Pirate 3 – Needs 1 extra vote


Can buy vote from 5 for 1 gold
Distribution is {0,0,99,0,1}
Pirate 2 – Needs 1 extra vote

Can buy vote from 4 for 1 gold

Distribution is {0,99,0,1,0}
Slide 13 of 13
Pirate Game:
A Sample Complex Analysis
Work backwards
Pirate 4 – Needs 0 extra votes

Distribution is {0,0,0,100,0}
Pirate 3 – Needs 1 extra vote


Can buy vote from 5 for 1 gold
Distribution is {0,0,99,0,1}
Pirate 2 – Needs 1 extra vote


Can buy vote from 4 for 1 gold
Distribution is {0,99,0,1,0}
Pirate 1 – Needs 2 extra votes


Can buy vote from 3 and 5 for 1 each
Distribution is {98,0,1,0,1}
Slide 14 of 13
Pirate Game:
A Sample Complex Analysis
Work backwards
Pirate 4 – Needs 0 extra votes

Distribution is {0,0,0,100,0}
Pirate 3 – Needs 1 extra vote


Can buy vote from 5 for 1 gold
Distribution is {0,0,99,0,1}
Pirate 2 – Needs 1 extra vote


Can buy vote from 4 for 1 gold
Distribution is {0,99,0,1,0}
Pirate 1 – Needs 2 extra votes


Can buy vote from 3 and 5 for 1 each
Distribution is {98,0,1,0,1}
Final answer: {98,0,1,0,1}
Slide 15 of 13
The Stag Hunt
Conflict modeling benefits of social cooperation: suggested by
Rousseau as a model for the social contract
Each player can choose to hunt a hare or a stag. If he hunts a
stag, he must have the cooperation of his partner in order to
succeed. An individual can get a hare by himself, but a hare is
worth less than a stag
Let’s examine a payoff matrix!
Slide 16 of 13
Stag Hunt: The Payoff
Stag
Hare
Stag
A, a
C, b
Hare
B, c
D, d
Stag
Hare
Stag
4,4
1,3
Hare
3,1
3,3
• Let a>b>or=d>c
•While each player choosing a hare
would represent a Nash Equilibrium, it
would not represent the maximum payoff
•Interestingly, the greatest payoff results
from mutual cooperation between the
players. This requires, however, an
element of trust between members of
society.
Slide 17 of 13