Game theory

Game Theory
Sinem Bulkan
PhD in Organisational Behaviour

Outline
O Introduction: Founders of the Game Theory
O Introduction: Application Areas in Branches of Science
O Glossary of Terms in Theory
O Game Theory Defined
O Assumptions of the Theory
O Types of Games
O Nash Equilibrium
O Zero-Sum and Non-Zero Sum Games
O Finite and Infinite Games
O Prisoner’s Dilemma and Tit-For-Tat Strategy
O Social Dilemmas
O Cooperation
O Co-Opetition
O References for Further Reading

John von Neumann (1903-1957)
O Hungarian-born American (Jewish)
O Received his PhD in Mathematics at the age of 22.
O Gottinghen University – Rockfeller scholarship
O First mastery paper ‘Mathematical Formulation of Quantum
Mechanics’ published at the age of 23.
O Invited to Princeton University in 1930 he remained there as a
mathematics professor until his death.
O During the war worked for Military, (MANIAC, NORC)
O He was included in the target selection committee responsible for
choosing the Japanese cities of Hiroshima and Nagasaki as the
first targets of the atomic bomb.
O Assigned to United Atomic Atom Energy Program in 1954.
O 150 published papers in his life; 60 in pure mathematics, 20 in
physics, and 60 in applied mathematics.
O Speculated that the reason for his death is a cancer due to the
nuclear tests.

John von Neumann (1903-1957)
O The inspiration for game theory
was poker.
O Poker was not only guided by
Probability Theory but also
Bluffing.
O Wanted to formalize the idea of
‘bluffing’ , a strategy that is meant
to deceive the other players and
hide information from them.
O John von Neumann & Oskar
Morgenstern (1944) Theory of
Games and Economic Behaviour

John Nash (1928)
• American Mathematician
• His works of game theory, differential geometry, and
partial differential equations have provided insight into the
forces that govern chance and events inside complex
systems in daily life.
• In 1978, Nash was awarded the John von Neumann
Theory Prize for his discovery of non-cooperative equilibria,
now called Nash equilibria.
• The PhD Thesis ‘Game Theory’ he published at the age of
21 enabled him to be awarded with Nobel Memorial Prize in
Economic Sciences in 1994. (27 pages!!!)
• Paranoid schizophrenia
• Creator of the games of “Hex” and “So Long Sucker”
• Now lives in New Jersey.

What is Game Theory? (The Math of
Competition)
O The study of mathematical models of conflict and cooperation
between intelligent rational decision-makers – (Myerson,2007)
O Game theoretic concepts apply when the actions of players are
interdependent. Concepts of the theory provide a language to
to formulate, structure, understand and analyze strategic
scenarios (by Turocy and Stengel, 2001)
O It arises whenever 2 or more individuals, with different values
or goals, compete to try to control the course of events.
O Uses mathematical tools, called games, to study situations that
involve both conflict and cooperation.

Application Areas
O In the 1950s and 1960s, game theory was broadened theoretically
and applied to problems of war and politics.
O Since the 1970s, it has driven a revolution in economic theory
(competitive markets, economic negotiations).
O Additionally, it has found applications in sociology and psychology,
and established links with evolution and biology.
O Game theory received special attention in 1994 with the awarding
of the Nobel prize in economics to Nash, John Harsanyi, and
Reinhard Selten.
O At the end of the 1990s, a high-profile application of game theory
has been the design of auctions.

4 Elements of Game Theory
O The players
• How many players are there?
• People, organisations, countries
O They choose from a list of options available to them, called
strategies.
O A complete description of the information available to
players at each decision made.
O A description of the consequences (payoffs) for each player
for every possible profile of strategy choices of all players.

Assumptions of Game Theory
O Payoffs are known and fixed.
O All players behave rationally.
 They understand and seek to maximize gains and
minimize losses.
 They are flawless in calculating which actions will
maximize their payoffs.
O The rules of the game are common knowledge:
 Each player knows the set of players, strategies and
payoffs from all possible combinations of strategies: call
this information ‘X’.
 Common Knowledge means that each player knows that
all players know X, that all players know that all players
know X, that all players know that all players know that all
players know X and so on,....., ad infinitium.

Outcome/Payoff
O The outcome depends on the choices of all the players.
O Many strategy choices are non-cooperative, such as those
between combatants in warfare or in sports.
O Some strategy choices are co-operative, aiming joint gains.
(economics and politics)
O Some applications are in bargaining tactics in labour-management
disputes, resource-allocation decisions in
political campaigns, military choices in international crises, and
the use of threats by animals in habitat acquisition and
protection.

Strategic Interaction – Game Example
O Players: x and y
O Strategies: Advertise or Not Advertise
O Payoffs: Companies’ Profits
O Strategic Landscape:
O Each firm earns $50 million from its customers
O Advertising costs a firm $20 million
O Advertising captures $30 million from competitor
O How to represent this game?

Representing a Game
PLAYERS
STRATEGIES
12
y
No Ad Ad
PAYOFFS
x
No Ad 50 , 50 20 , 60
Ad 60 , 20 30 , 30

What to Do?
If you are advising x, what strategy do you recommend?
O Best reply for x:
O If y advertises: Ad
O If y does not advertise: Ad
13
y
No Ad Ad
x
No Ad 50 , 50 20 , 60
Ad 60 , 20 30 , 30

Dominance (Baskın Strateji)
O A strategy for one player dominates another strategy if it
yields a higher payoff no matter what the other players
choose.
O Strategy U dominates strategy D for the player 1 below
You /
Opponent
L R
U 3,9 3,8
D 0,0 2,1
O A strategy is dominant if dominates all other strategies.
O No rational player would ever play a dominated
strategy, so if both players have dominant strategies,
we can make a unique prediction.

Nash Equilibrum (Nash Dengesi)
Ignore The Blonde – Go for the Brunettes
If everyone competes for the blond, we block each
other and no one gets her. So then we all go for her
friends. But they give us the cold shoulder,
because no one likes to be second choice. Again,
no winner. But what if none of us go for the blond.
We don’t get in each other’s way, we don’t insult
the other girls. That’s the only way to win. That’s
the only way we all get a girl (from A Beautiful
Mind)
Adam Smith ‘ the best result comes from everyone in the group doing what's best for
himself‘
John Nash ‘the best result will come where everyone in the group does what is best for
himself ... and the group.'

Nash Equilibrum (Nash Dengesi)
O Solution concept of a game involving two or more players, in which each player is
assumed to know the equilibrium strategies of the other players, and no player has
anything to gain by changing only his own strategy unilaterally.
O If each player has chosen a strategy and no player can benefit by changing strategies
while the other players keep theirs unchanged, then the current set of strategy
choices and the corresponding payoffs constitute a Nash equilibrium.
O if every player prefers not to switch (or is indifferent between switching and not) then
the set of strategies is a Nash equilibrium.
O Example: if Amy is making the best decision she can, taking into account Wili's
decision, and Wili is making the best decision he can, taking into account Amy's
decision. Likewise, a group of players are in Nash equilibrium if each one is making
the best decision that he or she can, taking into account the decisions of the others.

Nash Equilibrum (Nash
Dengesi)
Coordination Game
Player 2 adopts
strategy A
Player 2 adopts
strategy B
Player 1 adopts
strategy A
4,4 1,3
Player 1 adopts
strategy B
3,1 2,2

Prisoners’ Dilemma Game
(Tutsak İkilemi Oyunu)
O Two players, prisoners 1, 2. There is no physical evidence
to convict either one, so the prosecuter seeks a confession.
O Each prisoner has two strategies.
O Prisoner 1: Don’t Confess, Confess
O Prisoner 2: Don’t Confess, Confess
O Payoff consequences are quantified in prison years.
O More years=worse payoffs.
O Information about strategies and payoffs is complete; both
players (prisoners) know the available strategies and the
payoffs from the intersection of all strategies.
O Strategies are chosen by the two Prisoners simultaneously
and without communication.

Prisoners’ Dilemma Game
(Tutsak İkilemi Oyunu)
Prisoner 1
Prisoner 2
Don’t
Confess
Confess
Don’t
Confess
-1,-1 -15,0
Confess 0,-15 -5,-5
• Think of the payoffs as prison terms/years lost.
• Blue section shows the best strategy for both prisoners.

Can Communication Help?
O Suppose we recognize the Prisoner’s Dilemma and we can
talk to one another in advance, for instance, make
promises to not confess.
O If these promises are non-binding and / or there are little
consequences from breaking these promises (They are
‘cheap talk’) the the ability of the prisoners to communicate
prior to choosing their strategies may not matter.

TIT-FOR-TAT Strategy for Prisoners’
Dilemma
O Consists of just 2 simple maneuvers:
1. Cooperate on the first move.
2. On following moves, do whatever the other player did.
O Example: Defect, defect, cooperate, defect, cooperate
O Tit-for-that: Cooperate, defect, defect, cooperate, defect,
cooperate

TIT-FOR-TAT Strategy for Prisoners’
Dilemma
The strategy is distinguished by the following characteristics:
O It is nice. Those who employ the tit-for-tat strategy are never the
first to defect. This is critical to building a trustful environment, and
is the biggest predictor of success.
O It is provocable. When the other party defects, the first party
responds immediately and clearly.
O It is forgiving. If the other party backs off, then the first party
responds in kind. "Let bygones be bygones."
O It is simple. This is not computer chess. The tit-for-tat strategy
requires little strategic planning.

Prisoners’ Dilemma is an example of a
Non-Zero Sum Games
O 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.
O A game is non-zero sum, if players’ interests are not
always in conflict, so that there are opportunities for both to
gain.
O Example: When both players choose don’t confess in
Prisoners’ Dilemma, they both gain relative to both
choosing Confess.

Zero Sum Games (Sıfır Toplamlı Oyunlar)
O For every combination of strategies, the
total benefit to all players in the game
always adds to zero
O A player benefits at the expense of others
such as poker and chess

Zero Sum Games (Sıfır Toplamlı Oyunlar)
YOU/OPPONENT A B C
1 30, -30 -10,10 20,-20
2 10,-10 20,-20 -20,20

Non-Zero Sum Games
O Some outcomes have net results greater or
less than zero
O A gain by one player does not necessarily
correspond with a loss by another
You /
Opponent
A B
A 3,3 0,5
B 5,0 1,1

NON-ZERO SUM GAMES: BATTLE OF
THE SEXES ( a two player coordination
game)
WIFE
HUSBAND
OPERA FOOTBALL
OPERA 3,2 0,0
FOOTBALL 0,0 2,3

Chicken Dilemma Game
O In the prototypical chicken dilemma two rebellious teens
race towards each other in the car
O whoever swerves is considered a ‘chicken’ and loses.
O If neither turns away a bad crash happens and both loose

Chicken Dilemma Game
Swerve Straight
Swerve Tie, Tie Lose, win
Straight Win, Lose Crash, Crash
A payoff matrix of Chicken
Swerve Straight
Swerve 0, 0 -1, +1
Straight +1, -1 -10,-10
Chicken with numerical payoffs

Simultaneous-Sequential Move Games
O Games where players choose actions simultaneously are
simultaneous move games.
Examples: Prisoners’ Dilemma, Sealed-Bid Auctions
- Must acticipate what your opponent will do right now,
recognising that your oppenent is doing the same.
O Games where players choose actions in particular
sequence are sequential move games.
- Examples: Chess, Bargaining/Negotiations
- Must look ahead in order to know what action to choose now.

Finite-Infinite Games
Finite games
O definite beginning and ending
O played with the goal of winning
O Rules exist to ensure the game is finite
O Ex. receiving a degree from an educational institution
Infinite games
O Do not necessarliy have a knowable end point to reason
backward
O Repeated interaction continuse for infinite amount of time
O The rational long term behavior is affected by endless
moves of the players
O Ex. Repeated prisoner’s dilemma

Perfect Information-Imperfect
Information Games
O Players have perfect information if they know exactly what has
happened every time a decision needs to be made, e.g. in chess.
O Otherwise, the game is one of imperfect information
- Example: In the investment game, the sender and receiver
might be differentially informed about the investment outcome. For
example, the receiver may know that the amount invested is always
tripled, but the sender may not be aware of this fact.

Conflict and Game Theory
O Game theory is a good way of starting to analyse conflicts
O It sets out broad patterns of resource allocation
O Shows how conflict arises from the allocations
O Highlights potential solutions to those conflicts.

Conflict and Game Theory in a Daily
Life

Limitations to Game Theory
O Simple games can be modelled, such games are
rare in real life. This is the reason why books uses
the same examples.
O Most real conflicts involve multiple games and
multiple strategies.

‘In-Group Love’ and ‘Out-Group
Hate’ as Motives for Individual
Participation in Intergroup Conflict
Nir Halevy, Gary Bornstein, and Lilach Sagiv
The Hebrew University of Jerusalem
Association for Psychological Science Vol.19
No.4, 405-411
ARTICLE

Introduction
O Groups that fail to mobilize sufficient
participation will not survive the aggresion of
other groups,
and
O They will have to forgo the benefits of victory or,
worse yet, bear the costs of defeat.

Introduction
O Importance of solidarity mechanisms
O Groups with more effective means of instilling
self-sacrifice in their members have prevailed
over groups with less effective solidarity
mechanisms.

Modelling Intergroup Conflict
O Consideration of the internal tension between group
welfare and individual welfare.
O The relative success of the two groups in overcoming
this intragroup conflict determines the outcome of the
intergroup competition.
O A basic model of this two-level structure is the
intergroup prisoner’s dilemma game (IPD) game
(Bornstein, 1992, 2003; Bornstein & Ben Yossef,
1994)

Intergroup Prisoner’s Dilemma
(IPD) Game
O Game is played with 2 groups.
O 3 members in each group.
O Each player receives and endowment of 10 tokens
and can contribute any number of these tokens to the
group’s pool.
O For each token contributed by a member of the in-group,
each of its members, gaind 1 money unit
(MU), and each member of the out-group loses 1MU.
O For each token kept, the player is paid 2 MU.This
means contributing a token yields a return 1MU, but
at a cost of 2 MU.

Intergroup Prisoner’s Dilemma
(IPD) Game
O Dominant Individual Strategy: The strategy that yields the
highest personal payoffs regardless of what all the other in-group
and out-group members do – is to contribute
nothing.
O Dominant Group Strategy: The strategy that yields the
highes payoffs for each group regardless of what the other
group does – for all group members to contribute all their
tokens.
O Collectively Optimal Strategy: The strategy that maximizes
the payoff of all players in both groups – is for all players to
withhold contribution.

Intergroup Prisoner’s Dilemma –
Maximizing Difference (IPD-MD) Game
O The same game rules with IPD.
O However contributions can be made to two different pools.
O Contributing a token to the within-group pool (Pool W)
increases the payoff for each in-group member, by 1 MU,
without affecting the out-group. (a Cooperative motivation)
O Contributing a token to the between-group pool (Pool B)
increases the payoff for each in-group member, by 1 MU,
and at the same time decreases the payoff for each out-group
member. (aggresive motivation to hurt the out-group,
or a competitive motivation to increase the in-group’s
advantage over the out-group.

Intergroup Prisoner’s Dilemma –
Maximizing Difference (IPD-MD) Game
• Corner cells – IPD-MD
• Thick outline box – two independent 3-person PD game
• Triple outline box – Maximizing difference or ‘spite’ game

The Experiment: Group and Individual
Behaviour in the IPD and IPD-MD Games
O A laboratory experiment that compared behaviour in the
IPD and IPD-MD games.
O Made decisions without communicating with the other in-group
members (half the experimental sessions).
O Group member met for a short discussion before making
their decisions (other half).
O Communication increases cooperation by enhancing group
identity and commitment.

Method
O Participants
240 male students (mean age=24 years)
Promised monetary reward for participation.
O Design and Procedure
Cohorts of 6.
4 conditions (IPD or IPD MD with or without intragroup
communication).
Each cohort were divided into two 3-person groups.
Informed on rules and payoffs.
Pools were labeled ‘A’ and ‘B’.
Explained how each player’s own decision and the decision
of other players would affect the payoffs.

Method
O Design and Procedure
‘Decision Form’
10 tokens – Endowment, Pool W, Pool B
Pool W – 1 New Israeli Shekel (NIS) to each in-group member
Pool B – Subtracted 1 NIS from the payoff to each out-group
member
For each token a player kept, he was paid 2 NIS
Each player was paid a flat bonus of 30 NIS to endure that
payoffs would be positive
Participants could earn between 10NIS and 70 NIS.
Postexperimental questionnaire

Results
O Across the two communication conditions, the participants
contributed 63% of their endowment in the IPD-MD Game,
as compared with 51% in the IPD game.
O Within group communication increased the overall
contribution rate from 44% to 70%.
O Game type and communication did not have an interactive
effect on the overall contribution rate.

Results
O In the IPD MD game, contributions made almost
exclusively to the cooperative, within-group pool.
O Following communication, contributions to pool W
increased from 47% to 68%.
O In the IPD game, about 7% of the participants in the no-communication
condition contributed all 10 tokens to pool
B, and 33% contributed nothing.
O Following group discussion, 57% of the individuals
contributed everything but 18% contributed nothing.

Results
O In general, efficiency was higher in the IPD-MD.
O In both games, group communication enhanced individual
contributions.
O IPD-MD  Communication increased contributions to the
cooperative, within ingroup pool, thus enhancing efficiency.
O IPD Communication raised contributions to the
competitive, between- group pool, thereby diminishing
collective efficiency.

Results
O IPD-MD 6 players could have earned a total of 180 NIS if
they all had contributed their entire endowment to pool W.
O They actually earned about 78% of this amount in the no-communication
condition and almost 87% in the
communication condition.
O IPD 6 players could have earned a total of 120 NIS if
they had all kept their entire endowment.
O The average efficiency rate was 65% in the no-communication,
and only 32% in the communication
condition.

Discussion
O The peaceful group coexistence observed in the IPD-MD
game was utterly shattered in the IPD game, in which
maximizing the in-group’s gain was necessarily at the
expense of the out-group.
O Under these circumstances, group members did not
hesitate to compete.
O One obvious explanation for this behaviour is that the in-group
members placed more weight on the gains their
contribution produced for the in-group than on the loses it
inflicted on the out-group.

Implications for Conflict Resolution
O IPD-MD game suggest that intergroup conflicts can be
resolved by channeling group members’ alturism toward
internal group causes.
O Whereas in the IPD game, ‘peace’ is achieved only if all
members of both groups defect, in the IPD-MD game,
groups, can avoid war while maintaining their ability to
mobilize collective action.

Game theory

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