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Game Theory
Introduction to non-cooperative
games
Unit 3.
Plan
1. Non-cooperative game Theory Game concepts: Rules of the Game
Game concepts: Common Knowledge 2 players games
2. Strategic games Representation Solution concept
Dominant Strategy Nash Equilibrium
Pure strategies Mixed Strategies
3. Extensive Games Subgame perfect Nash Equilibrium
Learning Objectives
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1. Non-Cooperative Game Theory
A game (dictionary definition) is a A competitive activity in
which players contend with
each other according to a set of rules
Strategic Behavior:
Conscious behavior arising among competitors or players, in a
situation where all are aware of their
conflicting interests and interdependence of their
payoffs.
Non-Cooperative Game theory provides the tools and techniques to
analyze strategic
situations in which decision-makers interact.
A Game is a stylized model that depicts situations of strategic
behavior.
Every game is played by a set of rules which have to specify
four elements
Game Concepts: Rules of the Game
Who is playing: a set of players
What they are playing with- a set of alternative actions or
choices, the strategies, that each player has
available.
strategies (rules telling each player which action to choose at
each point of the game.
When each player gets to play
How much they stand to gain (or lose) from the choices made in
the game.
Payoffs (usually consist of the profit or expected profit the
players receive after the game have been played out)
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Game Concepts: Common Knowledge
We assume that the rules of the game are Common Knowledge
who, what, when, and how much constitutes the rules of the
game.
That is, If you ask any two players in the game about the rules
of the game they will give you the
same answer.
Every player knows the rules, and knows that other players knows
that he knows the rules,,
ad infinitum.
2 players game
Player 1
(i) knows the rules of the game, and knows that player 2 knows
the rules of the game, also
(ii) Knows that player 2 knows that player 1 knows (i), also
(iii) knows that player 2 knows (ii)
Ad infinitum
This is important because if at some point player 1 knows that
he will never choose some actions (because it is not the
best available), he must know that player 2 knows that too.
Other information structures
Information structure of the game refer to what players know and
when they know it during the
game.
Perfect information: each player knows all the history of the
game before taking any action.
Games where players move simultaneously are games of imperfect
information.
Incomplete information: some players have more information than
others, at the beginning of the game.
Symmetric information: all players have exactly the same
information when each player moves.
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Mathematical representation
Two principal mathematical representations of a game:
Strategic (strategic) form of a game which consider situations
in which actions are chosen
one and for all.
Extensive form of a game which allows for the possibility that
plan may be revised as they are
carried out. (dynamic games)
STRATEGIC GAMES
Examples
Representation of the game
Player 1 and player 2
Player 1 available actions 1, 1
Player 2 2, 2
Payoffs depends on own action and the other players action.
These games are games of imperfect information because both
players move
simultaneously.
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For any game we need to know what actions will be chosen by the
players? We also take into account that the best action of any
given player depends, in general, on the other players actions.
The player must form a belief about the other players actions: when
choosing an action a player must have in mind the
actions the other players will choose.
When we analyze a game we assume that Each player (rational
decision maker) chooses the best
available action, and
Every players belief about other players actions is correct
Solution concept
A solution concept is a formal rule for predicting how a game
will be played.
Predictions (called "solutions), describe which strategies will
be adopted by players and, therefore, the result of the game.
Iterated dominance
The most commonly used solution concepts are equilibrium
concepts, Examples: Nash equilibrium.
In many cases different solution concepts result in more than
one solution. We may apply a refinement to eliminate
implausible
equilibria in richer games. Example: Backward induction,
Strategic Game: Representation
P2 2 2
P1
1
2(1,2)
2(1,2)
1(1,2)
1(1,2)
1
2(1,2)
2(1,2)
1(1,2)
1(1,2)
Players
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P2 2 2
P1
1
2(1,2)
2(1,2)
1(1,2)
1(1,2)
1
2(1,2)
2(1,2)
1(1,2)
1(1,2)
Players
Actions
P2 2 2
P1
1
2(1,2)
2(1,2)
1(1,2)
1(1,2)
1
2(1,2)
2(1,2)
1(1,2)
1(1,2)
Players
Actions
Payoffs
Alternatively
P2 2 2
P1
1 1
(1,2), 2
(1,2) 1
(1,2), 2
(1,2)
1 1
(1,2) , 2
(1,2)
1(1,2)
, 2(1,2)
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Zero Sum Games
Games in which one players gain is always matched by another
players loss.
Typical example is the matching pennies game
Such games are called Strictly Competitive
Head Tail
Head 1, 1 1, 1
Tail 1, 1 1, 1
+ =
Sales of Ice Cream
Beginning Middle End
B 50,50 25,75 50, 50
M 75,25 50,50 75,25
E 50,50 25,75 50,50
Information Structures
Game of imperfect information
Both players move simultaneously to pick a location
Game of symmetric information
Both players have the same information when they play
Also, this is a zero-sum game
Because total sales (100%) will be divided between the two
players any gain for a player is exactly
marched by a loss for the other player.
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We are looking for the solution(s) for this game
We need to introduce the concepts of
Dominant strategies
Nash Equilibrium
Mini-max strategy
Dominant Strategy
A dominant strategy is a strategy that outperform any other
strategy no matter
what strategy the opponent selects.
Beginning Middle End
B 50,50 25,75 50, 50
M 75,25 50,50 75,25
E 50,50 25,75 50,50
Dominant Strategy
A dominant strategy is a strategy that outperform any other
strategy no matter
what strategy the opponent selects.
Beginning Middle End
B 50,50 25,75 50, 50
M 75,25 50,50 75,25
E 50,50 25,75 50,50
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Dominant Strategy
A dominant strategy is a strategy that outperform any other
strategy no matter
what strategy the opponent selects.
Beginning Middle End
B 50,50 25,75 50, 50
M 75,25 , 75,25
E 50,50 25,75 50,50
Equilibrium
Each player will play his dominant strategy and the solution
is:
, the payoff attached with this equilibrium is 50% of the sales
for each.
Iterated Dominance
In some games after eliminating a dominated strategy, one (or
more) other strategies
become dominated in their turn.
A rational player will proceed the same way and eliminate them
too.
The process of elimination of dominated strategies is called
iterated dominance.
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Iterated Dominance
Iterated Dominance
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Nash Equilibrium
Definition: A Nash equilibrium is an action profile with the
property that no player can do better by choosing an action
different from
, given that every other player adheres to .
Recall the Ice cream trucks game: the solution obtained by
eliminating strictly dominated action is also a Nash
Equilibrium.
Consider the strategy profile , player 1 has no incentive to
change and play Beginning
or End given that player 2 chooses Middle.
the same is true for player 2.
Which is not the case for any other action profile.
Consider whether the strategy profile , is a Nash
Equilibrium
Player 1 has incentive to play instead Middle and increase his
market share to 75%. (the same is true for
player 2).
Hence the strategy profile , is not a Nash equilibrium.
You might do the same with any other possible strategy profiles
in this game. How many are they?
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Matching Pennies Game
Two players each must secretly turn a penny to heads or tails.
The players reveal their
choices simultaneously. If the penny match
player1 keeps both pennies. If not player 2
keeps both. Head Tail
Head +1, 1 1, +1
Tail 1, +1 +1, 1
Matching Pennies Game
Consider {H,H}. Player 1 would like to change and play T instead
of H, if player 2 adhere to H.
Because at least one player can do better by choosing T, {H,H}
is not an equilibrium.
We can check that there is no equilibrium using the strategies T
or H (with certainty).
We say there is no equilibrium in pure strategies for this
game
But Nash Theorem says that there is at least one equilibrium for
any finite game!
Mixed Strategies
A mixed strategy is an assignment of a probability to each pure
strategy.
Since probabilities are continuous, there is an infinitely many
mixed strategies available to each player.
A player rationally picks the probabilities that maximize his
Expected payoff.
In the marching Pennies game let player 1 plays Head with
probability p, and player 2 plays head with probability q, then the
expected payoffs functions are the following:
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1 = +1 + 1 1 + 1
1 + 1 +1
After simplifications
1 = 2 1 + 1 + 2 1= 2 1 2 1
Provided an interior solution exists, expected profit is
maximized if 1
= 2 2 1 = 0, =
1
2
1-p
p
1-q q
2 = 1 + 1 1 + 1
+1 + 1 1
After simplifications
2 = 2 1 2 1
Provided an interior solution exists, expected profit is
maximized if 2
= 2 2 1 = 0, =
1
2
1-p
p
1-q q
Notice that the expected profit for player 1, if player 2 plays
the
mixed strategy q = , is
1 = 2 1 21
2 1 = 0, for any value of p. Hence the
value of p is irrelevant: whether player 1 uses H or T all the
time or
any of the two randomly (at any p different from ) his
expected
payoff will not be different, as long as the opponent uses a
mixed
strategy q=1/2.
The same is true for player 2, if player 1 plays the mixed
strategy p =
, his expected payoff will be always 2 = 2 1
21
2 1 = 0, for any value of q.
This is a characteristic of optimal mixed strategies.
1-p
p
1-q q
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(intuitively) optimal mixed strategy makes your opponents
selection of a strategy irrelevant to its
outcome.
We can use this intuition to solve for a mixed strategy without
using calculus:
Player 1: the optimal strategy is the one (p) that result in
equal payoff for player 2 whether heplays
always H or T
+ 1 = 1 1 2 = 2 1
2 = 4, = 1/2
1-p
p
1-q q
Similarly, we can prove that the optimal mixed strategy for
player 2 is q=1/2.
It is clear that strategy , = (1
2,
1
2) is a
Nash Equilibrium since no player can improve
on the outcome by playing a different strategy.
Notice that a pure strategy is a special case of mixed
strategies (as it is a degenerate mixed
strategy (attaching a probability 1 to one of
the available strategies).
Any equilibrium using a combination of strategies in some
proportion is called a
mixed strategy Nash Equilibrium.
A mixed strategy Nash Equilibrium is unstable because payoffs
are the same regardless of the
strategy adopted by the players, so they have
little incentives to maintain the equilibrium.
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Ice Cream Trucks Game again
Another concept of solution for Zero-Sum games is the mini-max
strategy: each player
chooses the strategy that minimizes the
maximum possible outcome for other player.
Prisoners Dilemma Games
Two suspects in a major crime are held in separate cells.
(players)
There is no enough evidence to convict either of them unless one
of them acts as an informer against the other.
Players: two suspects
Actions: Each players set of actions is {Quiet, Fink}
Preferences : Suspect 1s ordering from best to
This a game of imperfect information because both suspects move
simultaneously.
It is also a static game (because both suspects move
simultaneously)
Prisoners Dilemma
Solution:
1. Check whether there is a dominant strategy
Fink is a dominant strategy for Suspect 1(no matter what
strategy suspect 2 adopts) because 3>2 and 1>0. Likewise,
Fink is a dominant strategy for suspect 2.
The equilibrium is {Fink, Fink} and the payoffs (1,1).
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We can check that the strategy profile {Quiet, Quiet} is not a
Nash Equilibrium. (because at least one of the suspects may improve
on the payoff of 2 by playing (another strategy, i.e. Fink)
Upon checking the payoff matrix it is clear that if both
suspects cooperate (i.e. choose
{Quiet, Quiet}), they will be both better of
than with the no-cooperation outcome .
Prisoners Dilemma and Coordination
Upon checking the payoff matrix it is clear that if both
suspects cooperate (i.e. choose
{Quiet, Quiet}), they will be both better of
than with the no-cooperation outcome .
Read the Press
This game has been used to model the problem of deployment of
taxi cab over the
territory of Grand Tunis.
Le problme du service de taxis individuels dans le Grand-Tunis -
Les
jaunes...partout et nulle part (Premire partie)
La Presse de Tunisie April 18, 2013.
http://www.lapresse.tn/05022015/65903/le-probleme-du-service-de-taxis-individuels-dans-le-grand-
tunis.html
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EXTENSIVE GAMES
Many games are sequential, in which player 1 moves, then player
2 respond, then player 1 respond to player 2s responseThese games
are dynamic (as opposed to static).
They are represented by game trees. The tree is defined by nodes
and branches.
Nodes represent places where something happens in the game (such
as a decision by one of the players), and
branches indicate the various actions that players can
choose.
We represent nodes by solid circles and branches by arrows
connecting the nodes.
A properly constructed tree is called an extensive-form
representation
Example: Price competition
Perfect information
Find Nash Equilibria.
The rule look ahead and reason back.
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Player 2 optimal strategy
(L,L) because 2>1 and >0
Player 1 figure that out so if he plays H he earn 0, and if he
plays L he earns million,
hence he plays L
The (unique) Nash Equilibrium is (L,(L,L))
Imperfect information
The two firms typically select their prices simultaneously and
independently.
Neither firms observes the others move before making its
choice.
CREDIBLE THREATSSUBGAME
PERFECT NE
Example: Entry Game
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Summary
Game theory is the study of how interdependent decision makers
make choices.
A game must include players, actions, information, strategies,
payoffs, outcomes, and equilibria.
In simple zero-sum games, the mini-max strategy is a dominant
solution to the game.
At least one Nash Equilibrium (in pure or random strategies)
exists for all finite games.
Games comes with different information structures: Perfect
Information (every player knows the history of
play before taking any action)
In games with complete information, nature does not move first,
or natures move is observed by all
players.
In games of certain information, nature never moves after
another player moves.
In games of symmetric information, all players have exactly the
same information when each moves.
A dominant solution to the classical prisoners dilemma game
results in a non-optimal
solution for the players.