Lecture 1 - Introduction
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Introduction to Game Theory
Examples
Matrix form Games
Utility
Solution concepts
 Dominant Strategies
 Nash Equilibria
Complexity
Mechanism Design: reverse game theory
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The study of Game Theory in the context of
Computer Science, in order to reason about
problems from the perspective of computability
and algorithm design.
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Computing involves many different selfish
entities. Thus involves game theory.
The Internet, Intranet, etc.
◦ Many players (end-users, ISVs, Infrastructure
Providers)
◦ Players wish to maximize their own benefit and act
accordingly
◦ The trick is to design a system where it’s beneficial
for the player to follow the rules
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Theory
◦ Algorithm design
◦ Complexity
◦ Quality of game states (Equilibrium states in
particular)
◦ Study of dynamics
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Industry
◦ Sponsored search
◦ Other auctions
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Rational Player
◦ Prioritizes possible actions according to utility or
cost
◦ Strives to maximize utility or to minimize cost
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Competitive Environment
◦ More than one player at the same time
Game Theory analyzes how rational players
behave in competitive environments
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Matrix representation of the game
Thieves
honor
Defect
Thieves
honor
3,3
6,2
2<3
Defect
2,6
5,5
5<6
Row
Player
Column Player
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It is a dominant strategy to confess
A dominant strategy is a “solution concept”
Thieves
honor
Defect
Thieves
honor
3,3
6,2
Defect
2,6
5,5
6,10
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Internet Service Providers (ISP) often share
their physical networks for free
In some cases an ISP can either choose to
route traffic in its own network or via a
partner network
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ISP 1 needs to route traffic from s1 to t1
ISP 2 needs to route traffic from s2 to t2
The cost of routing
along each edge is
one
A
B
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ISP1 routes via B:
◦ Cost for ISP1: 1
◦ Cost for ISP2: 4
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B,A: s21 to
to tt12
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Cost matrix for the game:
ISP 2
ISP 1
A
B
A
3,3
6,2
B
2,6
5,5
Prisoners Dilemma Again
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The game consists of only one ‘turn’
All the players play simultaneously and are
unaware of what the other players do
Players are selfish, seek to maximize their own
benefit
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N = {1,…,n} players
Player i has actions
We will say “action” or “strategy”
The space of all possible action vectors is
A joint action is the vector a∈A
Player i has a utility function
If utility is negative we may call it cost
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A strategic game:
Players
Actions of each
player
Utility of each
player
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Action ai of player i is a weakly dominant
strategy if:
Action ai of player i is a strongly dominant
strategy if:
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Vilfredo Pareto
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An outcome a of a game is Pareto optimal if for
every other outcome b, some player will lose by
changing to b
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St. Petersburg Paradox:
◦ Toss a coin until tails, I pay
you
◦ What will you pay me to play?
“Utility of Money”, “Bernulli Utility”
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Preferences over lotteries
Completeness:
Transitivity:
Continuity:
Independence:
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Utility function over
lotteries, real valued,
expected utility
maximization
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Gamble A: 100%
Gamble B: 10%
89%
1%
€ 1,000,000
€ 5,000,000
€ 1,000,000
Nothing
Gamble C: 11%
89%
Gamble D: 10%
90%
€ 1,000,000
Nothing
€ 5,000,000
Nothing
Gamble A or B?
Experimental
”Fact”:
Gamble C or D?
Experimental
“Fact”:
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Gamble A: 100%
Gamble B: 10%
89%
1%
€ 1,000,000
€ 5,000,000
€ 1,000,000
Nothing
Gamble C: 11%
89%
Gamble D: 10%
90%
€ 1,000,000
Nothing
€ 5,000,000
Nothing
“Fact”:
“Fact”:
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VNM Axioms
Expected Utility Maximization
Mixed Nash Equilibrium exists
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Assume there’s a shared resource (network
bandwidth) and N players.
 Each player “uses” the common resource, by
choosing Xi from [0,1].
If
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Otherwise,
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Given that the other
players are fixed, what
Is the best response?
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This is an equilibrium
No player can improve
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The case for Privatization or central
control of commons
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A Nash Equilibrium is an outcome of the game
in which no player can improve its utility alone:
Alternative definition: every player’s action is a
best response:
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The payoff matrix:
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The payoff matrix:
Row player has
no incentive to
move up
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The payoff matrix:
Column player
has no incentive
to move left
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The payoff matrix:
So this is an
Equilibrium state
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The payoff matrix:
Same thing here
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2 players need to send a packet from point O to
the network.
They can send it via A (costs 1) or B (costs 2)
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The cost matrix:
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The cost matrix:
Equilibrium states
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2 players, each chooses Head or Tail
Row player wins if they match the column
player wins if they don’t
Utility matrix:
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2 players, each chooses Head or Tail
Row player wins if they match the column
player wins if they don’t
Utility matrix:
Row player is fine,
but Column player
wants to move left
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2 players, each chooses Head or Tail
Row player wins if they match the column
player wins if they don’t
Utility matrix:
Column player is
fine, but Row player
wants to move up
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2 players, each chooses Head or Tail
Row player wins if they match the column
player wins if they don’t
Utility matrix:
Row player is fine,
but Column player
wants to move right
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2 players, each chooses Head or Tail
Row player wins if they match the column
player wins if they don’t
Utility matrix:
Column player is fine,
but Row player wants
to move down
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2 players, each chooses Head or Tail
Row player wins if they match the column
player wins if they don’t
Utility matrix:
No equilibrium state!
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Players do not choose a pure strategy (one
specific strategy)
Players choose a distribution over their
possible pure strategies
For example: with probability p choose Heads,
and with probability 1-p choose Tails
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Row player chooses Heads with probability p
and Tails with probability 1-p
Column player chooses Heads with probability q
and Tails with probability 1-q
Row plays Heads:
Row plays Tails:
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Each player selects
where
is the
set of all possible distributions over Ai
An outcome of the game is the Joint Mixed
Strategy
An outcome of the game is a Mixed Nash
Equilibrium if for every player
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2nd definition of Mixed Nash Equilibrium:
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Definition:
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Definition:
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Property of Mixed Nash Equilibrium:
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No pure strategy Nash Equilibrium, only Mixed
Nash Equilibrium, for mixed strategy (1/3, 1/3,
1/3) .
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N ice cream vendors are spread on the beach
Assume that the beach is the line [0,1]
Each vendor chooses a location Xi, which
affects its utility (sales volume).
The utility for player i :
X0 = 0, Xn+1 = 1
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For N=2 we have a pure Nash Equilibrium:
0
1/2
1
No player wants to move since it will lose space
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For N=3 no pure Nash Equilibrium:
0
1/2
1
The player in the middle always wants to move to
improve its utility
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If instead of a line we will assume a circle, we
will always have a pure Nash Equilibrium where
every player is evenly distanced from each
other:
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N companies are producing the same product
Company I needs to choose its production
volume, xi ≥ 0
The price is determined based on the overall
production volume,
Each company has a production cost:
The utility of company i is:
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Case 1: Linear price,
no production cost
◦ Utility:
◦ Pure Nash Equilibrium is reached at:
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Case 2: Harmonic price, no production cost
◦ Company i’s utility:
◦ Companies have incentive to produce as much as they
can – no pure or mixed Nash Equilibrium
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n players wants to buy a single item which is on
sale
Each player has a valuation for the product,
Assume WLOG that
Each player submits its bid,
, all players
submit simultaneously.
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Case 1: First price auction
◦ The player with the highest bid wins
◦ The price equals the bid
◦ 1st Equilibrium is:
 The first player needs to know the valuation of the
second player – not practical
◦ 2nd Equilibrium is:
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Case 2: Second price auction: Vickrey Auction
◦ The player with the highest bid wins
◦ The price equals the second highest bid
 No incentive to bid higher than one’s valuation - a player’s
utility when it bids its valuation is at least as high than
when it bids any other value
 This mechanism encourages players to bid truthfully
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Mechanism Design: reverse game theory – set
up a game so that the equilibria has a desired
property
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no regret
correlated eq
mixed Nash
pure
Nash
bestresponse
dynamics
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a directed graph G = (V,E)
k source-destination pairs (s1 ,t1), …, (sk ,tk)
a rate (amount) ri of traffic from si to ti
for each edge e, a cost function ce(•)
◦ assumed nonnegative, continuous, nondecreasing
Example: (k,r=1)
s1
c(x)=x
c(x)=1
Flow = ½
t1
Flow = ½
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Traffic and Flows:
 fP = amount of traffic routed on si-ti path P
 flow vector f
routing of traffic
s
t
Selfish routing: what are the equilibria?
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Some assumptions:
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agents small relative to network (nonatomic game)
want to minimize cost of their path
Def: A flow is at Nash equilibrium (or is a Nash
flow) if all flow is routed on min-cost paths
[given current edge congestion]
Example:
s
x
1
Flow =
.5
t
Flow =
.5
s
x
1
Flow =
1
t
Flow =
0
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model, defn of Nash flows by [Wardrop 52]
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Nash flows exist, are (essentially) unique
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congestion game (payoffs fn of # of players)
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potential game (equilibria as optima)
◦ due to [Beckmann et al. 56]
◦ general nonatomic games: [Schmeidler 73]
◦ defined for atomic games by [Rosenthal 73]
◦ previous focus: Nash eq in pure strategies exist
◦ defined by [Monderer/Shapley 96]
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Def: the cost C(f) of flow f = sum of all costs
incurred by traffic (avg cost × traffic rate)
x
s
1
½
½
t
Cost = ½•½ +½•1 = ¾
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Def: the cost C(f) of flow f = sum of all costs
incurred by traffic (avg cost × traffic rate)
x
s
1
½
½
t
s
t
Cost = ½•½ +½•1 = ¾
Formally: if cP(f) = sum of costs of edges of P
(w.r.t. the flow f), then:
C(f) = P fP • cP(f)
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Note: Nash flows do not minimize the cost
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observed informally by [Pigou 1920]
x
s
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½
1
1
t
½ + 0•1 = 1
Cost of Nash flow = 1•1
Cost of optimal (min-cost) flow = ½•½ +½•1 = ¾
0
Price of anarchy := Nash/OPT ratio = 4/3
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Initial Network:
s
x
1
½
½
½
½
1
t
x
cost = 1.5
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Initial Network:
s
x
1
½
½
½
½
1
x
cost = 1.5
Augmented Network:
t
s
x
1
½
½
½ 0
½
1
t
x
Now what?
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Initial Network:
s
½
x
1
½
½
½
1
x
cost = 1.5
Augmented Network:
x
t
s
0
1
1
t
x
cost = 2
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Initial Network:
s
½
x
1
½
½
½
1
x
cost = 1.5
Augmented Network:
x
t
s
0
1
1
t
x
cost = 2
All traffic incurs more cost! [Braess 68]
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see also [Cohen/Horowitz 91], [Roughgarden 01]
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