Game Theory
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Game theory was developed by John Von
Neumann and Oscar Morgenstern in 1944 
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Economists!
One of the fundamental principles of game
theory, the idea of equilibrium strategies was
developed by John F. Nash, Jr. (A Beautiful
Mind), a Bluefield, WV native.
Game theory is a way of looking at a whole
range of human behaviors as a game.
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Components of a Game
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Games have the following
characteristics:
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Players
Rules
Payoffs
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Based on Information
Outcomes
Strategies
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Types of Games
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We classify games into several types.
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By the number of players:
By the Rules:
By the Payoff Structure:
By the Amount of Information Available to
the players
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Games as Defined by the
Number of Players:
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1-person (or game against nature,
game of chance)
2-person
n-person( 3-person & up)
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Games as Defined by the
Rules:
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These determine the number of
options/alternatives in the play of the
game.
The payoff matrix has a structure
(independent of value) that is a
function of the rules of the game.
Thus many games have a 2x2 structure
due to 2 alternatives for each player.
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Games as Defined by the
Payoff Structure:
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Zero-sum
Non-zero sum
(and occasionally Constant sum)
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Examples:
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Zero-sum
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Non-zero sum
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Classic games: Chess, checkers, tennis, poker.
Political Games: Elections, War , Duels ?
Classic games: Football (?), D&D, Video games
Political Games: Policy Process
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Games defined by information
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In games of perfect information, each player
moves sequentially, and knows all previous
moves by the opponent.
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Chess & checkers are perfect information games
Poker is not
In a game of complete information, the rules are
known from the beginning, along with all possible
payoffs, but not necessarily chance moves
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Strategies
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We also classify the strategies that we
employ:
It is natural to suppose that one player
will attempt to anticipate what the other
player will do. Hence
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Minimax - to minimize the maximum loss a defensive strategy
Maximin - to maximize the minimum gain an offensive strategy.
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Iterated Play
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Games can also have sequential play
which lends to more complex strategies.
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Tit-for-tat - always respond in kind.
Tat-for-tit - always respond conflictually to
cooperation and cooperatively towards
conflict.
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Game or Nash Equilibria
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Games also often have solutions or
equilibrium points.
These are outcomes which, owing to the
selection of particular reasonable
strategies will result in a determined
outcome.
An equilibrium is that point where it is not
to either players advantage to unilaterally
change
his or her mind.
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Saddle points
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The Nash equilibrium is also called a
saddle point because of the two curves
used to construct it:
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an upward arching Maximin gain curve
and a downward arc for minimum loss.
Draw in 3-d, this has the general shape of a
western saddle (or the shape of the universe;
and if you prefer). .
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Some Simple Examples
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Battle of the Bismark Sea
Prisoner’s Dilemma
Chicken
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The Battle of the Bismarck Sea
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Simple 2x2 Game
US WWII Battle
Japanese Options
Sail
Sail
North
South
US
Options
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Recon
North
2 Days
2 Days
Recon
South
1 Day
3 Days
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The Battle of the Bismarck Sea
Japanese Options
Sail
Sail
Minima
North
South of Rows
US
Options
Recon
North
2 Days
2 Days 2
Recon
South
1 Day
3 Days 1
Maxima of Columns 2
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The Battle of the Bismarck Sea
- examined
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This is an excellent example of a two-person
zero-sum game with a Nash equilibrium
point.
Each side has reason to employ a particular
strategy
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Maximin for US
Minimax for Japanese).
If both employ these strategies, then the
outcome will be Sail North/Watch North.
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Decision Tree
Decision Tree Version of Battle of Bismark Sea
Search
North
2
Sail North
Search
South
1
Japanese
Search
North
2
Sail South
Search
South
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The Prisoners Dilemma
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The Prisoner’s dilemma is also 2-person
game but not a zero-sum game.
It also has an equilibrium point, and
that is what makes it interesting.
The Prisoner's dilemma is best
interpreted via a “story.”
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A Simple Prisoner’s Dilemma
Prisoner A
~ Confess Confess
~ Confess
Prisoner B
-1
-1
Confess
-10
-10
0
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0
-5
-5
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Alternate Prisoner’s Dilemma
Language
Uses Cooperate instead of Confess to denote player cooperation with
each other instead of with prosecutor.
Prisoner A
Cooperate Defect
Cooperate
Prisoner B
-1
-1
Defect
-10
-10
0
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0
-5
-5
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What Characterizes a
Prisoner’s Dilemma
Uses Cooperate instead of Confess to denote player cooperation with
each other instead of with prosecutor.
Prisoner A
Cooperate Defect
Prisoner B
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Cooperate
Reward
Tempt
Reward
Sucker
Defect
Sucker
Punish
Tempt
Punish
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What makes a Game a
Prisoner’s Dilemma?
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We can characterize the set of choices in a
PD as:
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Temptation (desire to double-cross other player)
Reward (cooperate with other player)
Punishment (play it safe)
Sucker (the player who is double-crossed)
A game is a Prisoner’s Dilemma whenever:
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T>R>P>S
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Or Temptation > Reward > Punishment > Sucker
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What is the Outcome of a PD?
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The saddle point is where both Confess
This is the result of using a Minimax
strategy.
Two aspects of the game can make a
difference.
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The game assumes no communication
The strategies can be altered if there is
sufficient trust between the players.
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Solutions to PD?
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The Reward option is the joint optimal
payoff.
Can Prisoner’s reach this?
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Minimax strategies make this impossible
Are there other strategies?
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Iterated Play
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The PD is a single decision game in
which the Nash equilibrium results from
a dominant strategy.
In iterated play (a series of PDs),
conditional strategies can be selected
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Chicken
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The game that we call chicken is widely
played in everyday life
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bicycles
Cars
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James Dean – variant
Mad Max
Interpersonal relations
And more…
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The Game of Chicken
Driver A
~ Swerve
~ Swerve
Driver B
1
1
Swerve
2
4
4
2
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Swerve
3
3
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Chicken is an Unstable game
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There is no saddle point in the game.
No matter what the players choose, at
least one player can unilaterally change
for some advantage.
Chicken is therefore unstable.
We cannot predict the outcome
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Chicken is Nuclear Deterrence
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