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SCIT1003
Chapter 3: Prisoner’s Dilemma
Non-Zero Sum Game

Prof. Tsang
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Zero-Sum Games
• The sum of the payoffs remains constant
during the course of the game.
• Two sides in conflict, e.g. chess, sports
• Being well informed always helps a player
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Example of zero-sum game
Matching Pennies
Mis-matcher
matcher
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Rock-Paper-Scissors
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Military game: attack the easy or hard
pass?
Attacker
Easy pass
Easy pass
Hard pass
0.5, 0.5
0.4, 0.6
Defense
side
Hard pass 0,
Payoff is the winning probability.
1
1,
0
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Games of Conflict
• Two sides competing against each other
• Characteristics of zero-sum games: your loss
is my gain
• Simultaneous moves: lack of information
about the opponent’s move
• Logical circle of reasoning: I think that he
thinks that I think that …
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Zero-sum game matrices are sometimes
expressed with only one number in each box,
in which case each entry is interpreted as a
gain for row-player and a loss for columnplayer.
Player A
Player B
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Non-Zero Sum Game
Prisoner’s Dilemma
• A zero-sum game is one in which the players' interests
are in direct conflict, e.g. in football, one team wins and
the other loses; payoffs sum to zero.
• A game is non-zero-sum, if players interests are not
always in direct conflict, so that there are opportunities
for both to gain, e.g. games in economics
• For example, when both players choose Don't Confess in
the Prisoners' Dilemma
• Most game in reality have aspects of common interests
as well as conflict.
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Prisoners’ Dilemma: payoff matrix
Confess
Don’t
Confess
Confess
-5, -5
0, -10
Don’t
Confess
-10, 0
-3, -3
2
1
9
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Imperfect Information
• Partial or no information concerning the
opponent is given in advance to the player’s
decision, e.g. Prisoner’s Dilemma.
• Imperfect information may be diminished
over time if the same game with the same
opponent is played repeatedly.
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Games of Co-operation
Players may improve payoff through
• communicating
• forming binding coalitions & agreements
• do not apply to zero-sum games
Prisoner’s Dilemma
with Cooperation
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Strategies
• A strategy is a “complete plan of action” that fully
determines the player's behavior, a decision rule or set
of instructions about which actions a player should
take following all possible histories up to that stage.
• The strategy concept is sometimes (wrongly) confused
with that of a move. A move is an action taken by a
player at some point during the play of a game (e.g., in
chess, moving white's Bishop a2 to b3).
• A strategy on the other hand is a complete algorithm
for playing the game, telling a player what to do for
every possible situation throughout the game.
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Dominant or dominated strategy
• A strategy S for a player A is dominant if it
is always the best strategies for player A no
matter what strategies other players will
take.
• A strategy S for a player A is dominated if
there is at least a strategy better than it no
matter what strategies other players will
take.
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Rule: If you have a dominant strategy, use
it!
Use
strategy 1
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Dominance Solvable
COMMANDMENT
If you have a dominant strategy, use it.
Expect your opponent to use his/her dominant strategy
if he/she has one.
• If each player has a dominant strategy, the game is
dominance solvable
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Only one player has a Dominant
Strategy
Time
The Economist
G
S
S 100 , 100 0 , 90
G 95 , 100 95 , 90
• For The Economist:
– G dominant, S dominated
• Dominated Strategy:
• There exists another strategy which always does better regardless
of opponents’ actions
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How to recognize a Dominant Strategy
To determine if the row player has any dominant strategy
1.Underline the maximum payoff in each column
2.If the underlined numbers all appear in a row, then it is
the dominant strategy for the row player
No dominant strategy for the row player in this example.
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How to recognize a Dominant Strategy
To determine if the column player has any dominant strategy
1.Underline the maximum payoff in each row
2.If the underlined numbers all appear in a column, then it is the
dominant strategy for the column player
There is a dominant strategy for the column player in this example.
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If there is no dominant strategy
• Does any player have a dominant strategy?
• If there is none, ask “Does any player have a
dominated strategy?”
• If yes, then
• Eliminate the dominated strategies
• Reduce the normal-form game
• Iterate the above procedure
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Eliminate any dominated strategy
Eliminate
strategy 2 as
it’s dominated
by strategy 1
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Successive Elimination of Dominated
Strategies
• If a strategy is dominated, eliminate it
• The size and complexity of the game is
reduced
• Eliminate any dominated strategies from the
reduced game
• Continue doing so successively
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Example: Two competing Bars
•
•
•
•
Two bars (bar 1, bar 2) compete each other
Can charge price of $2, $4, or $5 for a drink
6000 tourists pick a bar randomly
4000 natives select the lowest price bar
No dominant strategy for the both players.
Bar 2
$2
$4
$5
$2 10 , 10 14 , 12 14 , 15
Bar 1 $4 12 , 14 20 , 20 28 , 15
$5 15 , 14 15 , 28 25 , 25
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Successive Elimination of Dominated
Strategies
$2
Bar 1
$2
$4
$5
Bar 2
$4
$5
10 , 10 14 , 12 14 , 15
12 , 14 20 , 20 28 , 15
15 , 14 15 , 28 25 , 25
Bar 2
$4
$5
$4 20 , 20 28 , 15
Bar 1
$5 15 , 28 25 , 25
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An example for Successive Elimination of strictly dominated
strategies, or the process of iterated dominance
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Equilibrium
• The interaction of all players' strategies results in an
outcome that we call "equilibrium."
• Traditional applications of game theory attempt to find
equilibria in games.
• In an equilibrium, each player is playing the strategy
that is a "best response" to the strategies of the other
players. No one is likely to change his strategy given the
strategic choices of the others.
• Equilibrium is not:
• The best possible outcome. Equilibrium in the one-shot prisoners'
dilemma is for both players to confess.
• A situation where players always choose the same action.
Sometimes equilibrium will involve changing action choices (known
as a mixed strategy equilibrium).
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Definition: Nash Equilibrium
“If there is a set of strategies with the property
that no player can benefit by changing
his/her strategy while the other players keep
their strategies unchanged, then that set of
strategies and the corresponding payoffs
constitute the Nash Equilibrium.”
Source: http://www.lebow.drexel.edu/economics/mccain/game/game.html
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Nash equilibrium
• If each player has chosen a strategy and no
player can benefit by changing his/her
strategy while the other players keep theirs
unchanged, then the current set of strategy
choices and the corresponding payoffs
constitute a Nash equilibrium.
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Conditions for Nash equilibrium
• Each player is choosing a best response to
what he believes the other players will do.
• Each player’s beliefs are correct. The other
players are all doing what everyone else thinks
they are doing.
Assumptions:
Rational players
“Putting yourself in the other person’s shoes”
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Example: B B Lean vs Rainbow’s End
p.105
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p.124
Games with infinitely many strategies
Rainbow’s End’s price
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(U, L) is not a Nash equilibrium because Player 2 can gain by
deviating alone to R; …
You can cross out those that are not Nash equilibria.
Finding Nash equilibria: (a) with strike-outs; (b) with
underlinings
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Sometimes, there is no Nash
Equilibrium
L
C
R
U
0, 2
2, 3
4, 1
M
1, 1
3, 1
0, 2
D
0, 3
1, 0
5, 1
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Sometimes, there are more than one Nash
Equilibrium.
No strictly dominant strategies and no strictly dominated
strategies.
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Hunting game: multi-Nash Equilibria
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Hunting game with 2-Nash Equilibria
Barney’s choice
Stag
Fred’s
choice
Bison
3
Stag
4
0
0
0
Bison
0
4
3
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Battle of sexes
Barney’s choice
Ballet
Alice’s
choice
Football
3
Ballet
4
0
0
0
Football
0
4
3
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Prisoner’s Dilemma: finding the Nash equilibrium
Which is a Nash Equilibrium?
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Prisoner’s Dilemma :
Applications
• Relevant to:
– Nuclear arms races.
– Dispute Resolution and the decision to hire a
lawyer.
– Corruption/political contributions between
contractors and politicians.
• How do players escape this dilemma?
– Play repeatedly
– Find a way to ‘guarantee’ cooperation
– Change payoff structure
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Nuclear arms races
prisoner's dilemma in disguise
Two countries try to decide whether to build the nuclear bombs.
Is there a Nash Equilibrium?
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Cigarette Advertising
prisoner's dilemma in disguise
Two companies try to decide whether to run cigarette advertisement.
Reynolds
No Ad
Ad
Philip Morris
No Ad
Ad
50 , 50
20 , 60
60 , 20
30 , 30
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Price it higher? Lower?
prisoner's dilemma in disguise
p.69
B B Lean
Rainbow’s
80
80
72k , 72k
70
24k , 110k
70
110k , 24k
70k , 70k
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OPEC: example of a cartel
• Collaborating in price-fixing guarantees higher
profit to all industrial producers at the
expense of consumers.
• OPEC is the most notable cartel of petroleum
producing countries.
• Cheating is a problem in price-collusion.
• Governments in many nations have enacted
anti-trust laws (Sherman Act in the US) to
protect consumers. (p.90-94)
Fight together or run-away?
prisoner's dilemma in disguise
Your friend
Fight
you
Runaway
Fight
0.8 , 0.8
Run-away
0 ,
1
1 ,
0.1 , 0.1
0
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Collaborative behavior in biology
• Hunting packs of wolves
• Sharing between colony of vampire bats in
Rica Costa (p.98)
• Ants
• Bees
• Humans
Sustainability of resources sharing
• Community resources sharing is generally
viewed as a form of cooperative game similar
to Prisoner’s Dilemma by most people.
• However, over-exploitation & cheating in
conservation effort lead to depletion of
resources. Its consequence is much deeper
than the simple (& superficial) payoff matrix
would suggest. (p.94-97)
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Tragedy of the Commons
• When individuals, acting independently & rationally,
will deplete a shared common resource even when
doing so is not in their best interest.
• An example to explain overuse of shared resources.
• Extend the Prisoner’s Dilemma to more than two
players.
• Each member of a group of neighboring farmers
prefers to let his cow/sheep to graze on the commons,
rather than keeping it on his own inadequate land, but
the commons will be rendered unsuitable for grazing if
it is overgrazed.
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In the beginning, there is a nice piece of grassland
owned by all villagers. “What a waste!” said a farmer.
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Happy farmers with their well-fed cows.
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“Why not have more cows? Why waste the resource?”
said the farmers.
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In the end, sad farmers with their hungry cows.
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Tragedy of the Commons
an apparent payoff matrix at the start
Your neighbor
Add a cow
Don’t add
add
a cow 2
2
Don’t
add 1
2
1
2
1
1
As long as the common pasture is not overgrazed, adding one
more cow is the dominant strategy for everybody.
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Tragedy of the Commons
an apparent payoff matrix in between
Your neighbor
Add a cow
Don’t add
add
a cow 1.5
1.5
Don’t
add 0.9
1.8
0.9
1.8
1
1
When the common pasture is starting to be overgrazed, adding
cow is still the dominant strategy for everybody, but the return is
getting worse.
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Tragedy of the Commons
the final form of payoff matrix
Your neighbor
Add a cow
Don’t add
add
a cow 0.5
0.5
Don’t
add 0.7
1.1
0.7
1.1
1
1
Finally when the common pasture is overgrazed, adding cow is
no longer the dominant strategy for everybody.
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Tragedy of the Commons
• Problem: cost of maintenance is
ignored/externalized
– Farmers don’t adequately pay for their impact.
– Resources are overused due to inaccurate estimates of
cost.
• Relevant to:
– Health or other social benefits
– Environmental laws, overfishing, whaling, pollution, etc.
– Global warming
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Environmental policy
Tragedy of the Commons in disguise
Factory B
pollution
No
pollution
50 , 50
60 , 20
Factory A
pollution
No
pollution
20 , 60
20 , 20
Two factories producing same chemical can choose to
pollute (lower production cost) or not to pollute (higher
production cost).
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Global warming
Tragedy of the Commons in disguise
Emissions
Reduced?
No
Country A
Yes
Country B
No
Yes
50 , 50
60 , 20
20 , 60
20 , 20
Two countries producing CO2 can choose to reduce
(higher production cost) or not to reduce (lower
production cost).
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Another Example:
Big & Little Pigs
Cost to press
button = 2 units
When button is pressed,
food given = 10 units
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Decisions, decisions...
What’s the best strategy for the little pig? Does he have a dominant
strategy?
Does the big pig have a dominant strategy?
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Research in industries
Big & Little Pigs
in disguise Small Company
research
Big
Company
research
No
research
5 , 1
No
research
4 , 4
9 , -1
0 , 0
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Summary: Ch. 3
• A strategy is a “complete plan of action” that
fully determines the player's behavior.
• Dominant strategy is the best strategy no
matter what strategies other players will take.
• A dominated strategy is one at least there is a
strategy better than it no matter what strategies
other players will take.
• If you have a dominant strategy, use it!
• Eliminate any dominated strategy
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Assignment 3.1
Assignment 3.1
N-person Investment game
• In a group of N>>2 persons
• If you invest $0, you get $0 in return
• If you invest $10, you get
– $5 return, if more than 60% of the group invest
– -$10 return, if less than 60% of the group invest
• Is there any Nash equilibrium?
• Answer: 2,
• What are they? Why?
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Example in real life
• Which technology to choose?
– Window or Apple
– Betamax or VHS (video format in 1980s)
• What else??
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