A Crash Course in Game Theory
Werner Raub
Workshop on Social Theory, Trust, Social
Networks, and Social Capital II
National Chengchi University – NCCU
April 2011
Aim of the session
• A very brief introduction to game theory, introducing
in an intuitive fashion the tools (concepts,
assumptions, theorems) that we need for the
subsequent session on “Game-theoretic models for
effects of social embeddedness on trust and
cooperation” and that are likewise theoretical
background for many studies and applications that
we discuss during this workshop
Social situations with interdependent actors and
game theory as a tool for analyzing such situations
• Compare Max Weber’s famous definition of
sociology and of social action: “Sociology […] is a
science which attempts the interpretive
understanding of social action in order thereby to
arrive at a causal explanation of its course and
effects […] Action is social in so far as […] it takes
account of the behaviour of others and is thereby
oriented in its course.” (Weber 1974: 88; emphasis
added)
• Game theory: A set of tools (concepts, assumptions,
theorems) for modeling and analyzing social
situations with interdependent actors.
Game-theoretic models and the P-T-E scheme
P–T-E
•
•
Focus on “T” in P-T-E
Focus on the two (!) ingredients of “T”:
1. Theory / theoretical model
2. Testable hypotheses that are generated from
(ideally: deduced from) theory / theoretical
model plus additional assumptions
Game-theoretic models and Coleman’s scheme
Collective
effects: macrooutcomes such
as Pareto(sub)optimally
Social conditions:
interdependencies as
summarized in the
game description
Equilibrium behavior
Individual preferences
as summarized in the
game description
Individual strategy
choices and
equilibrium path
behavior
Game theory: further reading – textbooks
Prisoner’s Dilemma
Player 2
C
D
C
R,R
S,T
D
T,S
P,P
Player 1
Assumptions:
• T>R>P>S
• Simultaneous moves
• Binding agreements are not feasible (“noncooperative game”)
• Information: each player is informed on his or her own alternative actions
and outcomes, as well as on alternative actions and outcomes for the
partner
An intuitive characterization of
'goal-directed' action
Actors have:
• Alternative actions
• Goals, i.e., they evaluate the possible outcomes of
their actions
• Expectations (or information) on the “states of the
world” (for example, expectations on certain
“contingencies” or on the behavior of other actors).
• Assumption: actors choose the action that seems
most appropriate, given their expectations, to realize
their goals.
Goal-directed action in interdependent situations:
some core concepts of game theory I
• Consider a noncooperative game between two players A
and B (generalizing the definitions for the case of more
than two players is straightforward). Let X be a strategy
of actor A and let Y be a strategy of actor B.
• X is a best reply strategy of player A against strategy Y of
player B if X maximizes A’s payoff against Y.
– Note: Using strategy X against Y is consistent with
goal-directed behavior of A, given that A anticipates B
to use Y.
• X is a dominant strategy of player A if X is player A’s
unique best reply against all strategies of player B.
– Note: Goal-directed behavior implies that a player
uses a dominant strategy.
– Note: A player has at most one dominant strategy and
often he or she has none.
Goal-directed action in interdependent situations:
some core concepts of game theory II
•
A strategy combination (X, Y) is a
Nash equilibrium if X is a best reply strategy of
player A against Y and Y is a best reply strategy of
player B against X.
– Note: Given that A anticipates B to use Y and B
anticipates A to use X, playing Nash equilibrium
is consistent with goal-directed behavior.
– Note: Nash has shown that every “finite game”
has at least one equilibrium, possibly in mixed
strategies.
– Note: A game often has more than one
equilibrium.
John Nash
On John Nash: A Beautiful Mind
Goal-directed action in interdependent situations:
some hypotheses of game theory
• H1: A player chooses a best reply strategy, given his
or her anticipation of the strategy chosen by the
other player
• H2: If a player has a dominant strategy, he or she will
use this strategy
• H3: The chosen strategies will be a Nash equilibrium
Goal-directed action in interdependent situations:
some core concepts of game theory III
• A strategy combination (X, Y) is
Pareto-optimal if there is no other strategy
combination that yields higher payoffs for at least 1
player, and not lower for the other player.
• A strategy combination (X, Y) is
Pareto-suboptimal if there is another strategy
combination that yields higher payoffs for at least 1
player, and not lower for the other player.
Application to the (non repeated)
Prisoner’s Dilemma
• D is a dominant strategy.
• Hence, (D,D) is the unique Nash equilibrium.
• Hence, (D,D) will be played according to H2 as well
as H3.
• (D,D) is Pareto-suboptimal.
• (C,C) is Pareto-optimal and better for both players
than (D,D).
• But: (C,C) is not a Nash equilibrium!