Game Theory

Game Theory
Bernhard Nebel, Ingo Thon, Robert Mattmüller,
Malte Helmert, Tim Schulte, Dominik Winterer
Lecture Notes
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Contents
1 Introduction
1.1 What is Game Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Areas of Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
1
2 Strategic Games
2.1 Dominated Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Iterative Elimination of Strictly Dominated Strategies . . . . . . . . .
4
5
6
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1 Introduction
1.1 What is Game Theory?
Game Theory is the analysis of strategic decision situations where multiple players interact
with each other.
The result of a game depends on the decisions of the involved players and all players are aware
of this fact. This raises the question of what the outcome of the game will be assuming each
player acts rationally, i.e. tries to maximize its own utility, and each player itself assumes
that all other players act rationally as well.
Originally, game theory was a research field of theoretical economics but has come to play an
increasingly important role in artificial intelligence and computer science, especially when
modelling distributed, heterogeneous systems of selfish agents. While solution concepts for
games are already known, many algorithmic questions remain to be answered. Since the
assumption of players’ rationality is more reasonable for artificial agents than for natural
agents, game theory might even be more applicable to computer science than to economics.
Example 1 (Beauty contest). Every player chooses a natural number between 1 and 100.
The players that come closest to 2/3 of the average win.
Example 2 (Congestion games). Consider a network of streets with travel costs dependent
on the number of agents choosing a particular road. The goal of an agent is to travel from
s to t minimizing travel cost. Which route will the agents take when there are n = 2, 3, . . .
agents? What is the average travel cost per agent?
n1/n
1
s
t
0
1
n2/n
1.2 Areas of Game Theory
One area of game theory deals with the analysis of strategic games (normal form games)
where each player selects a strategy from a predefined set of strategies and the outcome of
the game is determined by the resulting strategy combination.
1
1 Introduction
Example 3 (Prisoner’s dilemma). Two members of a criminal gang are arrested and interrogated separately. The prosecutors lack sufficient evidence to convict the pair on the accused
crime. Each prisoner is given the opportunity to defect and betray the other prisoner or to
cooperate with the other by remaining silent. Possible outcomes are:
1. both cooperate and each of them is sentenced to three months of prison,
2. only one prisoner defects and gets out of prison while the cooperating prisoner is
sentenced to ten years of prison,
3. both defect and each of them is sentenced to three years of prison.
Another area of game theory regards extensive-form games where players perform multiple moves (e.g. chess or repeated strategic games). We differentiate between games with
perfect information, like chess, and games with imperfect information, like poker.
Coalition or negotiation games model situations such as negotiations, distribution of
payoffs, or elections.
Mechanism design considers games from the perspective of a game designer who specifies
the rules of the game with the intention to optimize the social welfare of all players.
The most important solution concepts in game theory are the elimination of dominated
strategies, Nash equilibria, and other equilibria notions.
2
1 Introduction
3
2 Strategic Games
Definition 4 (Strategic game). A strategic game G = hN, (Ai ), (ui )i consists of
• a finite nonempty set of players N ,
• for each player i ∈ N , a nonempty set Ai of actions/strategies and
• for each player i ∈ N , payoff function ui : A → R, where
Y
A=
Ai .
i∈N
The strategic game G is finite if A is finite.
Instead of payoff functions, we will sometimes use the notion of preference relations. The
preference relation i for player i ∈ N is defined as follows:
a i b iff ui (a) ≥ ui (b).
We can model a finite strategic game using a payoff matrix. As an example, consider a
two-player game where each player has two actions. player 1 has actions T and B, player 2
has actions L and R.
player 2
L
R
T
w1 , w2
x1 , x2
B
y1 , y2
z1 , z2
player 1
If player 1 chooses action T and player 2 chooses action L, then player 1 gets payoff w1 and
player 2 gets payoff w2 , i. e. in every cell of the payoff matrix the first value is the payoff of
player 1, and the second is the payoff of player 2.
Example 5 (Prisoner’s dilemma). Consider the prisoner’s dilemma as described above.
Let D be the strategy to defect and C to cooperate with the fellow prisoner. The payoffs
correspond to the months in prison:
C
D
C
−3, −3
−120, 0
D
0, −120
−36, −36
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2 Strategic Games
Example 6 (Hawk and dove). In a fight for resources, two players can behave either like
a hawk (H) or like a dove (D). If both players behave like doves, both players share the
benefit, if a hawk meets a dove, the hawk wins and gets the bigger part. However if two
hawks meet each other, then the benefit gets lost completely because they will fight each
other.
D
H
D
3, 3
1, 4
H
4, 1
0, 0
Example 7 (Matching pennies). Two players can choose either heads (H) or tails (T ) of a
coin. If both players make the same choice, then player 1 receives one Euro from player 1,
and if they make different choices, then player 2 gets one Euro from player 1.
H
T
H
1, −1
−1, 1
T
−1, 1
1, −1
Example 8 (Bach or Stravinsky). Two persons, one of whom prefers Bach whereas the
other one prefers Strawinsky want to go to a concert together. For both it is more important
to go to the same concert than to go to their favorite concert. Let B be the action of going
to the Bach concert, S the action of going to the Strawinsky concert.
Strawinsky enthusiast
B
S
B
2, 1
0, 0
S
0, 0
1, 2
Bach enthusiast
2.1 Dominated Strategies
Q
Notation 9. Let a = (ai )i∈N be strategy profile (a ∈ A = i∈N Ai ). Furthermore we
define a−i := (aj )j∈N \{i} and (a−i , ai ) = (aj )j∈(N \{i})∪{i} = (aj )j∈N = a.
Definition 10 (Strictly dominated strategy). An action ai ∈ Ai in strategic game hN, (Ai ), (ui )i
is called strictly dominated if there is an action a+
i ∈ Ai such that for all a ∈ A:
ui (a−i , ai ) < ui (a−i , a+
i ).
5
2 Strategic Games
Remark 11. It is never rational to play strictly dominated strategies.
2.1.1 Iterative Elimination of Strictly Dominated Strategies
• cancel out strategies that are strictly dominated as long as possible.
• if there is only one strategy profile left then that is the solution.
Example 12. Firstly, strike out the first row which is strictly dominated by the second
row. Then eliminate the second column which is strictly dominated by the first column.
C
D
C
3,3
0,4
D
4,0
1,1
D
C
D
4,0
1,1
Example 13. Iterative elimination of strictly dominated strategies in three steps:
L
R
T
2,1
0,0
M
1,2
2,1
B
0,0
1,1
L
R
L
T
2,1
0,0
T
2,1
M
1,2
2,1
M
1,2
Remark 14. Only rarely there is strict dominances between actions.
Remark 15. The result of iterative elimination of strict dominance is unique, i. e., independent of the elimination order.
Definition 16 (Weakly dominated strategies). An action ai ∈ Ai in game hN, (Ai ), (ui )i is
called weakly dominated, if there is an action a+
i ∈ Ai such that for all a ∈ A
ui (a−i , ai ) ≤ ui (a−i , a+
i )
and for at least one a ∈ A, we have
ui (a−i , ai ) < ui (a−i , a+
i ).
Example 17. The result of iterative elimination of weakly dominated strategies is in general
not unique and depends on the elimination order. To see this, consider the following game:
6
2 Strategic Games
L
R
T
2, 1
0, 0
M
2, 1
1, 1
B
0, 0
1, 1
First of all, we eliminate action T (T ≤ M ), then we eliminate action L (L ≤ R). As a result
every possible strategy profile, i.e. M R or BR, in the reduced game has payoff profile (1, 1).
An alternative to this elimination order is to first eliminate B (B ≤ M ), and then R (R ≤ L)
leading to a reduced game with strategy profiles T L and M L. Both strategy profiles have
payoff profiles (2, 1) and give player 1 –unlike the other ordering–a higher utility compared
to the reduced game where only strategy profiles M R and BR remained.
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2 Strategic Games
8
Bibliography
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9
Index
Aktion, 4, 31
Allokation, 62
approximative, 64
Allokationsproblem für single-minded bidders, 63
Anreizkompatibilität
einer sozialen Entscheidungsfunktion,
53
eines Mechanismus, 58
eines Mechanismus für single-minded
bidders, 64
Approximation
einer Allokation, 64
Arrows Unmöglichkeitssatz, 50, 51
Auktion
Vickrey-Auktion, 57
Auktionen
kombinatorische, 62
Auszahlungsfunktion, 4, 31
Einstimmigkeit
partielle, 50
totale, 50
Ergebnis, 33
Erweiterung einer sozialen Entscheidungsfunktion, 54
Erweiterungslemma, 54
Evolutionäres Gleichgewicht, 21
Gefangenendilemma, 4
Gemischte Erweiterung, 16
Gleichgewicht
teilspielperfektes, 35
Greedy-Mechanismus für single-minded bidders, 65
Hawk und dove, 5
Historie, 31
Horizont
endlicher, 31
Bach or Stravinsky, 5
Berechnungseffizienz
eines Mechanismus für single-minded
bidders, 64
Bewertung
Kombinatorische Auktionen, 62
single-minded, 63
Borda-Wahlverfahren, 46
Individuelle Rationalität, 59
Kakutani’s fixed point theorem, 17
Kemeny-Young-Methode, 49
Kombinatorische Auktionen, 62
kritische Bezahlungen, 65
Kuchenverteilspiel, 39
Linear Complementarity Problem, 26
Lineares Programm, 25
Cauchy-Schwarzsche Ungleichung, 67
Clarke-Pivot-Funktion, 59
Clarke-Pivot-Regel, 59
Condorcet-Methoden, 48
Condorcet-Paradoxon, 46
Matching Pennies, 5
Maximinimierer, 11
Mechanismus, 58
für single-minded bidders, 64
Vickrey-Clarke-Groves-Mechanismus,
58
Mechanismusdesign, 57
Mehrheitsentscheidung, 46
Monotonie
Diktator
soziale Entscheidungsfunktion, 53
soziale Wohlfahrtsfunktion, 50
Dominanz
schwache, 6
strikte, 5
10
INDEX
einer sozialen Entscheidungsfunktion,
53
eines Mechanismus für single-minded
bidders, 65
symmetrisches strategisches, 10
Spieler, 4, 31
Strategie
evolutionär stabile, 21
gemischte, 15
in extensiven Spielen mit perfekter Information, 32
reine, 4
Strategische Manipulation, 53
strategy profile, 5
Nash-Gleichgewicht, 7
in gemischten Strategien, 16
Neutralität
paarweise, 51
Nullsummenspiel, 10
Nutzen
erwarteter, 16
Teilspiel, 34
Teilspielperfektes Gleichgewicht, 35
Top-Präferenz, 54
Transfer
positiver, 59
Paarweise Neutralität, 51
Pluralitätsverfahren, 46
Pluralitätswahl, 47
in zwei Runden, 47
positiver Transfer, 59
Präferenzrelation, 46
Präferenzwahl mit übertragbaren Stimmen,
47
Unabhängigkeit von irrelevanten Alternativen, 50
Unterstützungsmenge, 16
Vickrey-Auktion, 57
Vickrey-Clarke-Groves-Mechanismus, 58
Satz
von Arrow, 51
von Conitzer und Sandholm, 28
von Gibbard-Satterthwaite, 55
von Kakutani, 17
von Kuhn, 37
von Nash, 16
von Vickrey-Clarke-Groves, 58
Schritt-Abweichung, 35
Schulze-Methode, 48
Single-minded
Bewertung, 63
bid, 63
bidder, 63
Soziale Effizienz, 62
Soziale Entscheidung, 46
Soziale Entscheidungsfunktion, 46
Soziale Wohlfahrt
einer Allokation, 62
Soziale Wohlfahrtsfunktion, 46
Spiel
extensives mit perfekter Information,
31
und simultanen Zügen, 39
und Zufallszügen, 38
strategisches, 4
Wahlverfahren
Condorcet-Methoden, 48
Kemeny-Young-Methode, 49
Pluralitätswahl, 47
in zwei Runden, 47
Präferenzwahl mit übertragbaren Stimmen, 47
Schulze-Methode, 48
11