IA168 Algorithmic Game Theory
Tomáš Brázdil
1
Organization of This Course
Sources:
Lectures (slides, notes)
based on several sources
slides are prepared for lectures, some stuff on greenboard
(⇒ attend the lectures)
Books:
Nisan/Roughgarden/Tardos/Vazirani, Algorithmic Game
Theory, Cambridge University, 2007.
Available online for free:
http://www.cambridge.org/journals/nisan/downloads/Nisan_Non-printable.pdf
Tadelis, Game Theory: An Introduction, Princeton
University Press, 2013
(I use various resources, so please, attend the lectures)
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Evaluation
Oral exam
Homework
(occasionally)
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What is Algorithmic Game Theory?
First, what is the game theory?
According to the Oxford dictionary it is "the branch of mathematics
concerned with the analysis of strategies for dealing with competitive
situations where the outcome of a participant’s choice of action
depends critically on the actions of other participants"
According to Myerson it is "the study of
mathematical models of conflict and cooperation
between intelligent rational decision-makers"
What does the "algorithmic" mean?
It means that we are "concerned with the computational
questions that arise in game theory, and that enlighten game
theory. In particular, questions about finding efficient algorithms
to ‘solve’ games.”
Let’s have a look at some examples ....
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Prisoner’s Dilemma
Two suspects of a serious crime are
arrested and imprisoned.
Police has enough evidence of only
petty theft, and to nail the suspects for
the serious crime they need testimony
from at least one of them.
The suspects are interrogated
separately without any possibility of
communication.
Each of the suspects is offered a deal:
If he confesses (C) to the crime, he is
free to go. The alternative is not to
confess, that is remain silent (S).
Sentence depends on the behavior of both suspects.
The problem: What would the suspects do?
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Prisoner’s Dilemma – Solution(?)
C S
C −5, −5 0, −20
S −20, 0 −1, −1
Rational "row" suspect (or his adviser) may reason as follows:
If my colleague chooses C, then playing C gives me −5 and
playing S gives −20.
If my colleague chooses S, then playing C gives me 0 and
playing S gives −1.
In both cases C is clearly better (it strictly dominates the other
strategy). If the other suspect’s reasoning is the same, both choose C
and get 5 years sentence.
Where is the dilemma? There is a solution (S, S) which is better for
both players but needs some “central” authority to control the players.
Are there always “dominant” strategies?
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Nash equilibria – Battle of Sexes
A couple agreed to meet this evening, but cannot
recall if they will be attending the opera or a football
match.
The husband would like to go to the football game.
The wife would like to go to the opera. Both would
prefer to go to the same place rather than different
ones.
If they cannot communicate, where should they go?
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Nash equilibria – Battle of Sexes
Battle of Sexes can be modeled as a game of two players (Wife,
Husband) with the following payoffs:
O F
O 2, 1 0, 0
F 0, 0 1, 2
Apparently, no strategy of any player is dominant. A “solution”?
Note that whenever both players play O, then neither of them wants
to unilaterally deviate from his strategy!
(O, O) is an example of a Nash equilibrium (as is (F, F))
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Mixed Equilibria – Rock-Paper-Scissors
R P S
R 0, 0 −1, 1 1, −1
P 1, −1 0, 0 −1, 1
S −1, 1 1, −1 0, 0
This is an example of zero-sum games: whatever one of the
players wins, the other one looses.
What is an optimal behavior here? Is there a Nash equilibrium?
Use mixed strategies: Each player plays each pure strategy with
probability 1/3. The expected payoff of each player is 0 (even if
one of the players changes his strategy, he still gets 0!).
How to algorithmically solve games in mixed strategies? (we
shall use probability theory and linear programming)
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Philosophical Issues in Games
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Games of Incomplete Information
In all previous games the players knew all details of the game
they played, and this fact was a “common knowledge”. This is
not always the case.
Example: Sealed Bid Auction
Two bidders are trying to purchase the same item.
The bidders simultaneously submit bids b1 and b2 and the item
is sold to the highest bidder at his bid price (first price auction)
The payoff of the player 1 (and similarly for player 2) is
calculated by
u1(b1, b2) =



v1 − b1 b1 > b2
1
2 (v1 − b1) b1 = b2
0 b1 < b2
Here v1 is the private value that player 1 assigns to the item and
so the player 2 does not know u1.
How to deal with such a game? Assume the “worst” private value?
What if we have a partial knowledge about the private values? 11
Mechanism Design
Suppose you are the game designer. How would
you design the game so that the “solutions” will
satisfy certain “global objectives” ?
Examples:
Sealed Bid Auctions: How would you design auction rules so
that for every bidder, bidding the private value will be a dominant
strategy?
How would you design protocols (such as network protocols), to
encourage “cooperation” (e.g., diminish congestion)?
This is an extremely hot topic of current research!
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Inefficiency of Equilibria
In Prisoner’s Dilemma, the selfish behavior
of suspects (the Nash equilibrium) results in
somewhat worse than ideal situation.
C S
C −5, −5 0, −20
S −20, 0 −1, −1
Defining a welfare function W which to every pair of strategies
assigns the sum of payoffs, we get W(C, C) = −10 but
W(S, S) = −2.
The ratio
W(C,C)
W(S,S) = 5 measures the inefficiency of "selfish-behavior"
(C, C) w.r.t. the optimal “centralized” solution.
Price of Anarchy is the maximum ratio between values of equilibria
and the value of an optimal solution.
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Inefficiency of Equilibria – Selfish Routing
Consider a transportation system where many
agents are trying to get from some initial location to
a destination. Consider the welfare to be the
average time for an agent to reach the destination.
There are two versions:
“Centralized”: A central authority tells each agent where to go.
“Decentralized”: Each agent selfishly minimizes his travel time.
Price of Anarchy measure the ratio between average travel time in
these two cases.
Problem: Bound the price of anarchy over all routing games?
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Dynamic Games
So far we have seen games in strategic form that are unable to
capture games that unfold over time (such as chess).
For such purpose we need to use extensive form games:
P1
P2
(1, 2)
C
(1, −1)
D
(0, 2)
E
A
P2
(2, 2)
F
(1, 3)
G
B
How to "solve" such games?
What is their relationship to the strategic form games?
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Chance and Imperfect Information
Some decisions in the game tree may be by chance and controlled by
neither player (e.g. Poker, Backgammon, etc.)
Sometimes a player may not be able to distinguish between several
“positions” because he does not know all the information in them
(Think a card game with opponent’s cards hidden).
F G
D 1
2
F G
E1
2
A
H I J
B
P1
P1
Nature
P2
(a, b) (c, d) (e, f) (g, h) (i, j) (k, ) (m, n)
Again, how to solve such games? 16
Games in Computer Science
Game theory is a core foundation of mathematical economics. But
what does it have to do with CS?
Games in AI: modeling of “rational” agents and their interactions.
Games in Algorithms: several game theoretic problems have
a very interesting algorithmic status and are solved by
interesting algorithms
Games in modeling and analysis of reactive systems: program
inputs viewed “adversarially”, bisimulation games, etc.
Games in computational complexity: Many complexity classes
are definable in terms of games: PSPACE, polynomial hierarchy,
etc.
Games in Logic: modal and temporal logics,
Ehrenfeucht-Fraisse games, etc.
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Games in Computer Science
Games, the Internet and E-commerce: An extremely active
research area at the intersection of CS and Economics
Basic idea: “The internet is a HUGE experiment in interaction
between agents (both human and automated)”
How do we set up the rules of this game to harness “socially
optimal” results?
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Summary and Brief Overview
This is a theoretical course aimed at some fundamental results of
game theory, often related to computer science
We start with strategic form games (such as the Prisoner’s
dilemma), investigate several solution concepts (dominance,
equilibria) and related algorithms (in particular, Lemke-Howson
algorithm for computing Nash Eq.)
Subsequently, we move on to incomplete information games,
auctions, and mechanism design
Then consider (in)efficiency of equilibria (such as the Price of
Anarchy) and its properties on important classes of routing and
network formation games.
Then we consider repeated games which allow players to learn
from history and/or to react to deviations of the other players.
Remaining time will be devoted to selected topics from extensive
form games, games on graphs etc.
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Static Games of Complete Information
Strategic-Form Games
Solution concepts
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Static Games of Complete Information – Intuition
Proceed in two steps:
1. Each player simultaneously and independently chooses
a strategy. This means that players play without observing
strategies chosen by other players.
2. Conditional on the players’ strategies, payoffs are distributed to
all players.
Complete information means that the following is common knowledge
among players:
all possible strategies of all players,
what payoff is assigned to each combination of strategies.
Definition 1
A fact E is a common knowledge among players {1, . . . , n} if for every
sequence i1, . . . , ik ∈ {1, . . . , n} we have that i1 knows that i2 knows
that ... ik−1 knows that ik knows E.
The goal of each player is to maximize his payoff (and this fact is
common knowledge).
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Strategic-Form Games
To formally represent static games of complete information we define
strategic-form games.
Definition 2
A game in strategic-form (or normal-form) is an ordered triple
G = (N, (Si)i∈N , (ui)i∈N), in which:
N = {1, 2, . . . , n} is a finite set of players.
Si is a set of (pure) strategies of player i, for every i ∈ N.
A strategy profile is a vector of strategies of all players
(s1, . . . , sn) ∈ S1 × · · · × Sn.
We denote the set of all strategy profiles by S = S1 × · · · × Sn.
ui : S → R is a function associating each strategy profile
s = (s1, . . . , sn) ∈ S with the payoff ui(s) to player i, for every
player i ∈ N.
Definition 3
A zero-sum game G is one in which for all s = (s1, . . . , sn) ∈ S we
have u1(s) + u2(s) + · · · + un(s) = 0. 22
Example: Prisoner’s Dilemma
N = {1, 2}
S1 = S2 = {S, C}
u1, u2 are defined as follows:
u1(C, C) = −5, u1(C, S) = 0, u1(S, C) = −20,
u1(S, S) = −1
u2(C, C) = −5, u2(C, S) = −20, u2(S, C) = 0,
u2(S, S) = −1
(Is it zero sum?)
We usually write payoffs in the following form:
C S
C −5, −5 0, −20
S −20, 0 −1, −1
or as two matrices:
C S
C −5 0
S −20 −1
C S
C −5 −20
S 0 −1
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Example: Cournot Duopoly
Two identical firms, players 1 and 2, produce some good.
Denote by q1 and q2 quantities produced by firms 1 and 2, resp.
The total quantity of products in the market is q1 + q2.
The price of each item is κ − q1 − q2 (here κ is a positive
constant)
Firms 1 and 2 have per item production costs c1 and c2, resp.
Question: How these firms are going to behave?
We may model the situation using a strategic-form game.
Strategic-form game model (N, (Si)i∈N , (ui)i∈N)
N = {1, 2}
Si = [0, ∞)
u1(q1, q2) = q1(κ − q1 − q2) − q1c1
u2(q1, q2) = q2(κ − q1 − q2) − q2c2
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Solution Concepts
A solution concept is a method of analyzing games with the objective
of restricting the set of all possible outcomes to those that are more
reasonable than others.
We will use term equilibrium for any one of the strategy profiles that
emerges as one of the solution concepts’ predictions.
(I follow the approach of Steven Tadelis here, it is not completely standard)
Example 4
Nash equilibrium is a solution concept. That is, we “solve” games by
finding Nash equilibria and declare them to be reasonable outcomes.
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Assumptions
Throughout the lecture we assume that:
1. Players are rational: a rational player is one who chooses his
strategy to maximize his payoff.
2. Players are intelligent: An intelligent player knows everything
about the game (actions and payoffs) and can make any
inferences about the situation that we can make
3. Common knowledge: The fact that players are rational and
intelligent is a common knowledge among them.
4. Self-enforcement: Any prediction (or equilibrium) of a solution
concept must be self-enforcing.
Here 4. implies non-cooperative game theory: Each player is in
control of his actions, and he will stick to an action only if he finds it to
be in his best interest.
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Evaluating Solution Concepts
In order to evaluate our theory as a methodological tool we use the
following criteria:
1. Existence (i.e. How often does it apply?): Solution concept
should apply to a wide variety of games.
E.g. We prove that mixed Nash equilibria exist in all two player finite
strategic-form games.
2. Uniqueness (How much does it restrict behavior?): We demand
our solution concept to restrict the behavior as much as possible.
E.g. So called strictly dominant strategy equilibria are always unique as
opposed to Nash eq.
The basic notion for evaluating "social outcome" is the following
Definition 5
A strategy profile s ∈ S Pareto dominates a strategy profile s ∈ S if
ui(s) ≥ ui(s ) for all i ∈ N, and ui(s) > ui(s ) for at least one i ∈ N.
A strategy profile s ∈ S is Pareto optimal if it is not Pareto dominated
by any other strategy profile.
We will see more measures of social outcome later.
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Solution Concepts – Pure Strategies
We will consider the following solution concepts:
strict dominant strategy equilibrium
iterated elimination of strictly dominated strategies (IESDS)
rationalizability
Nash equilibria
For now, let us concentrate on
pure strategies only!
I.e., no mixed strategies are allowed. We will generalize to
mixed setting later.
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Notation
Let N = {1, . . . , n} be a finite set and for each i ∈ N let Xi be
a set. Let X := i∈N Xi = {(x1, . . . , xn) | xj ∈ Xj, j ∈ N}.
For i ∈ N we define X−i := j i Xj, i.e.,
X−i = {(x1, . . . , xi−1, xi+1, . . . , xn) | xj ∈ Xj, ∀j i}
An element of X−i will be denoted by
x−i = (x1, . . . , xi−1, xi+1, . . . , xn)
We slightly abuse notation and write (xi, x−i) to denote
(x1, . . . , xi, . . . , xn) ∈ X.
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Strict Dominance in Pure Strategies
Definition 6
Let si, si
∈ Si be strategies of player i. Then si
is strictly
dominated by si (write si si
) if for any possible combination of
the other players’ strategies, s−i ∈ S−i, we have
ui(si, s−i) > ui(si , s−i) for all s−i ∈ S−i
Claim 1
An intelligent and rational player will never play a strictly
dominated strategy.
Clearly, intelligence implies that the player should recognize dominated
strategies, rationality implies that the player will avoid playing them.
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Strictly Dominant Strategy Equilibrium in Pure Str.
Definition 7
si ∈ Si is strictly dominant if every other pure strategy of player i is
strictly dominated by si.
Observe that every player has at most one strictly dominant strategy,
and that strictly dominant strategies do not have to exist.
Claim 2
Any rational player will play the strictly dominant strategy (if it exists).
Definition 8
A strategy profile s ∈ S is a strictly dominant strategy equilibrium if
si ∈ Si is strictly dominant for all i ∈ N.
Corollary 9
If the strictly dominant strategy equilibrium exists, it is unique and
rational players will play it.
Is the strictly dominant strategy equilibrium always Pareto optimal?
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Examples
In the Prisoner’s dilemma:
C S
C −5, −5 0, −20
S −20, 0 −1, −1
(C, C) is the strictly dominant strategy equilibrium (the only
profile that is not Pareto optimal!).
In the Battle of Sexes:
O F
O 2, 1 0, 0
F 0, 0 1, 2
no strictly dominant strategies exist.
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Indiana Jones and the Last Crusade
(Taken from Dixit & Nalebuff’s "The Art of Strategy" and a lecture of Robert
Marks)
Indiana Jones, his father, and the Nazis have all converged at the site
of the Holy Grail. The two Joneses refuse to help the Nazis reach the
last step. So the Nazis shoot Indiana’s dad. Only the healing power of
the Holy Grail can save the senior Dr. Jones from his mortal wound.
Suitably motivated, Indiana leads the way to the Holy Grail. But there
is one final challenge. He must choose between literally scores of
chalices, only one of which is the cup of Christ. While the right cup
brings eternal life, the wrong choice is fatal. The Nazi leader
impatiently chooses a beautiful gold chalice, drinks the holy water,
and dies from the sudden death that follows from the wrong choice.
Indiana picks a wooden chalice, the cup of a carpenter. Exclaiming
"There’s only one way to find out" he dips the chalice into the font and
drinks what he hopes is the cup of life. Upon discovering that he has
chosen wisely, Indiana brings the cup to his father and the water
heals the mortal wound. 33
Indiana Jones and the Last Crusade (cont.)
Indy Goofed
Although this scene adds excitement, it is somewhat
embarrassing that such a distinguished professor as Dr. Indiana
Jones would overlook his dominant strategy.
He should have given the water to his father without testing it
first.
If Indiana has chosen the right cup, his father is still saved.
If Indiana has chosen the wrong cup, then his father dies
but Indiana is spared.
Testing the cup before giving it to his father doesn’t help, since if
Indiana has made the wrong choice, there is no second chance
– Indiana dies from the water and his father dies from the wound.
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