MATH 1020 Chapter 1: Introduction to Game theory

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MATH 1020Chapter 1 Introduction to Game theory

• Dr. Tsang

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Why do we like games?

• amusement, thrill and the hope to win
• uncertainty course and result of a game

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Reasons for uncertainty

• randomness
• combinatorial multiplicity
• imperfect information

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Three types of games
bridge
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Game Theory ???

• Game theory is the study of how people interact
  and make decisions.
• This broad definition applies to most of the
  social sciences, but game theory applies
  mathematical models to this interaction under the
  assumption that each person's behavior impacts
  the well-being of all other participants in the
  game. These models are often quite simplified
  abstractions of real-world interactions.

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A cultural comment

• The Chinese translation ??? may be a bit
  misleading.
• Games are serious stuffs in western culture.
• The Great Game the strategic rivalry and
  conflict between the British Empire and the
  Russian Empire for supremacy in Central Asia
  (1813-1907).
• Wargaming informal name for military
  simulations, in which theories/tactics of warfare
  can be tested and refined without the need for
  actual hostilities.

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What does game mean?

• an activity engaged in for diversion or amusement
• a procedure or strategy for gaining an end
• a physical or mental competition conducted
  according to rules with the participants in
  direct opposition to each other
• a division of a larger contest
• any activity undertaken or regarded as a contest
  involving rivalry, strategy, or struggle ltthe
  dating gamegt ltthe game of politicsgt
• animals under pursuit or taken in hunting

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The Great Game
Political cartoon depicting the Afghan Emir Sher
Ali with his "friends" the Russian Bear and
British Lion (1878)
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What is Game Theory?

• Game theory is a study of how to mathematically
  determine the best strategy for given conditions
  in order to optimize the outcome
• how rational individuals make decisions when
  they are aware that their actions affect each
  other and when each individual takes this into
  account

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Brief History of Game Theory

• Game theoretic notions go back thousands of years
  (Sun Tzus writings????)
• 1913 - E. Zermelo provides the first theorem of
  game theory asserts that chess is strictly
  determined
• 1928 - John von Neumann proves the minimax
  theorem
• 1944 - John von Neumann Oskar Morgenstern write
  "Theory of Games and Economic Behavior
• 1950-1953 - John Nash describes Nash equilibrium
  (Nobel price 1994)

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Rationality

• Assumptions
• humans are rational beings
• humans always seek the best alternative in a set
  of possible choices
• Why assume rationality?
• narrow down the range of possibilities
• predictability

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Utility Theory

• Utility Theory based on
• rationality
• maximization of utility
• may not be a linear function of income or wealth

Utility is a quantification of a person's
preferences with respect to certain behavior as
oppose to other possible ones.
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Game Theory in the Real World

• Economists
• innovated antitrust policy
• auctions of radio spectrum licenses for cell
  phone
• program that matches medical residents to
  hospitals.
• Computer scientists
• new software algorithms and routing protocols
• Game AI
• Military strategists
• nuclear policy and notions of strategic
  deterrence.
• Sports coaching staffs
• run versus pass or pitch fast balls versus
  sliders.
• Biologists
• what species have the greatest likelihood of
  extinction.

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What are the Games in Game Theory?

• For Game Theory, our focus is on games where
• There are 2 or more players.
• There is some choice of action where strategy
  matters.
• The game has one or more outcomes, e.g. someone
  wins, someone loses.
• The outcome depends on the strategies chosen by
  all players there is strategic interaction.
• What does this rule out?
• Games of pure chance, e.g. lotteries, slot
  machines. (Strategies don't matter).
• Games without strategic interaction between
  players, e.g. Solitaire.

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Game Theory

• Finding acceptable, if not optimal, strategies in
  conflict situations.
• An abstraction of real complex situation
• Assumes all human interactions can be understood
  and navigated by presumptions
• players are interdependent
• uncertainty opponents actions are not entirely
  predictable
• players take actions to maximize their
  gain/utilities

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Types of games

• zero-sum or non-zero-sum if the total payoff of
  the players is always 0
• cooperative or non-cooperative if players can
  communicate with each other
• complete or incomplete information if all the
  players know the same information
• two-person or n-person
• Sequential vs. Simultaneous moves
• Single Play vs. Iterated

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Essential Elements of a Game

• The players
• how many players are there?
• does nature/chance play a role?
• A complete description of what the players can do
  the set of all possible actions.
• The information that players have available when
  choosing their actions
• A description of the payoff consequences for each
  player for every possible combination of actions
  chosen by all players playing the game.
• A description of all players preferences over
  payoffs.

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Normal Form Representation of Games

• A common way of representing games, especially
  simultaneous games, is the normal form
  representation, which uses a table structure
  called a payoff matrix to represent the available
  strategies (or actions) and the payoffs.

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A payoff matrix to Ad or not to Ad
PLAYERS
Philip Morris Philip Morris
No Ad Ad
Reynolds No Ad 50 , 50 20 , 60
Reynolds Ad 60 , 20 30 , 30
STRATEGIES
PAYOFFS
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The Prisoners' Dilemma????

• Two players, prisoners 1, 2.
• Each prisoner has two possible actions.
• Prisoner 1 Don't Confess, Confess
• Prisoner 2 Don't Confess, Confess
• Players choose actions simultaneously without
  knowing the action chosen by the other.
• Payoff consequences quantified in prison years.
• If neither confesses, each gets 3 year
• If both confess, each gets 5 years
• If 1 confesses, he goes free and other gets 10
  years
• Prisoner 1 payoff first, followed by prisoner 2
  payoff
• Payoffs are negative, it is the years of loss of
  freedom

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Prisoners Dilemma payoff matrix
Confess Dont Confess
Confess -5, -5 0, -10
Dont Confess -10, 0 -3, -3
2
1
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Prisoners Dilemma Example of Non-Zero Sum Game

• 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
  payoffs sum to zero.
• A game is non-zero-sum, if players interests are
  not always in direct conflict, so that there are
  opportunities for both to gain.
• For example, when both players choose Don't
  Confess in the Prisoners' Dilemma

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Zero-Sum Games

• The sum of the payoffs remains constant during
  the course of the game.
• Two sides in conflict
• Being well informed always helps a player

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Non-zero Sum Game

• The sum of payoffs is not constant during the
  course of game play.
• Some nonzero-sum games are positive sum and some
  are negative sum
• Players may co-operate or compete.

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Information

• Players have perfect information if they know
  exactly what has happened every time a decision
  needs to be made, e.g. in Chess.
• Otherwise, the game is one of imperfect
  information.

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Imperfect Information

• Partial or no information concerning the opponent
  is given in advance to the players decision,
  e.g. Prisoners Dilemma.
• Imperfect information may be diminished over time
  if the same game with the same opponent is to be
  repeated.

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Games of Perfect Information

• The information concerning an opponents move is
  well known in advance, e.g. chess.
• All sequential move games are of this type.

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Games of Co-operation

• Players may improve payoff through
• communicating
• forming binding coalitions agreements
• do not apply to zero-sum games
• Prisoners Dilemma
• with Cooperation

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Games of Conflict

• Two sides competing against each other
• Usually caused by complete lack of information
  about the opponent or the game
• Characteristic of zero-sum games

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Example of zero-sum game
Matching Pennies
matcher
Mis-matcher
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Rock-Paper-Scissors
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Zero-sum game matrices are sometimes expressed
with only one number in each box, in which case
each entry is interpreted as a gain for
row-player and a loss for column-player.
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Strategies

• A strategy is a complete plan of action that
  fully determines the player's behavior, a
  decision rule or set of instructions about which
  actions a player should take following all
  possible histories up to that stage.
• The strategy concept is sometimes (wrongly)
  confused with that of a move. A move is an action
  taken by a player at some point during the play
  of a game (e.g., in chess, moving white's Bishop
  a2 to b3).
• A strategy on the other hand is a complete
  algorithm for playing the game, telling a player
  what to do for every possible situation
  throughout the game.

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Dominant or dominated strategy

• A strategy S for a player A is dominant if it is
  always the best strategies for player A no matter
  what strategies other players will take.
• A strategy S for a player A is dominated if it is
  always one of the worst strategies for player A
  no matter what strategies other players will take.

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If you have a dominant strategy, use it!
Opponent
Strategy 2
Strategy 1
150
1000
Strategy 1
You
25
Strategy 2
- 10
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Dominance Solvable

• If each player has a dominant strategy, the game
  is dominance solvable

COMMANDMENT If you have a dominant strategy, use
it. Expect your opponent to use his/her dominant
strategy if he/she has one.
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Only one player has a Dominant Strategy
The Economist The Economist
G S
Time S 100 , 100 0 , 90
Time G 95 , 100 95 , 90

• For The Economist
• G dominant, S dominated
• Dominated Strategy
• There exists another strategy which always does
  better regardless of opponents actions

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How to recognize a Dominant Strategy

• To determine if the row player has any dominant
  strategy
• Underline the maximum payoff in each column
• If the underlined numbers all appear in a row,
  then it is the dominant strategy for the row
  player

No dominant strategy for the row player in this
example.
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How to recognize a Dominant Strategy

• To determine if the column player has any
  dominant strategy
• Underline the maximum payoff in each row
• If the underlined numbers all appear in a column,
  then it is the dominant strategy for the column
  player

There is a dominant strategy for the column
player in this example.
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If there is no dominant strategy

• Does any player have a dominant strategy?
• If there is none, ask Does any player have a
  dominated strategy?
• If yes, then
• Eliminate the dominated strategies
• Reduce the normal-form game
• Iterate the above procedure

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Eliminate any dominated strategy
Opponent
Strategy 2
Strategy 1
150
1000
Strategy 1
You
Strategy 2
25
- 10
-15
160
Strategy 3
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Successive Elimination of Dominated Strategies

• If a strategy is dominated, eliminate it
• The size and complexity of the game is reduced
• Eliminate any dominant strategies from the
  reduced game
• Continue doing so successively

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Example Two competing Bars

• Two bars (bar 1, bar 2) compete
• Can charge price of 2, 4, or 5 for a drink
• 6000 tourists pick a bar randomly
• 4000 natives select the lowest price bar

No dominant strategy for the both players.
Bar 2
2 4 5
Bar 1 2 10 , 10 14 , 12 14 , 15
Bar 1 4 12 , 14 20 , 20 28 , 15
Bar 1 5 15 , 14 15 , 28 25 , 25
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Successive Elimination of Dominated Strategies
Bar 2
5
4
2
2
14
,
15
14
,
12
10
,
10
,
,
,
4
Bar 1
Bar 1
28
,
15
20
,
20
12
,
14
,
,
,
5
25
,
25
15
,
28
15
,
14
,
,
,
4 5
Bar 1 4 20 , 20 28 , 15
Bar 1 5 15 , 28 25 , 25
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An example for Successive Elimination of strictly
dominated strategies, or the process of iterated
dominance
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Equilibrium

• The interaction of all players' strategies
  results in an outcome that we call "equilibrium."
  
• Traditional applications of game theory attempt
  to find equilibria in games.
• In an equilibrium, each player is playing the
  strategy that is a "best response" to the
  strategies of the other players. No one is likely
  to change his strategy given the strategic
  choices of the others.
• Equilibrium is not
• The best possible outcome. Equilibrium in the
  one-shot prisoners' dilemma is for both players
  to confess.
• A situation where players always choose the same
  action. Sometimes equilibrium will involve
  changing action choices (known as a mixed
  strategy equilibrium).

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Definition Nash Equilibrium

• If there is a set of strategies with the
  property that no player can benefit by changing
  his/her strategy while the other players keep
  their strategies unchanged, then that set of
  strategies and the corresponding payoffs
  constitute the Nash Equilibrium.
• Source http//www.lebow.drexel.edu/economics/mcca
  in/game/game.html

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Nash equilibrium

• If each player has chosen a strategy and no
  player can benefit by changing his/her strategy
  while the other players keep theirs unchanged,
  then the current set of strategy choices and the
  corresponding payoffs constitute a Nash
  equilibrium.

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No strictly dominant strategies and no strictly
dominated strategies.
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Finding Nash equilibria (a) with strike-outs
(b) with underlinings
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Prisoners Dilemma finding Dominated Strategies
Which is a Nash Equilibrium?
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Prisoners Dilemma Applications

• Relevant to
• Nuclear arms races.
• Dispute Resolution and the decision to hire a
  lawyer.
• Corruption/political contributions between
  contractors and politicians.
• How do players escape this dilemma?
• Play repeatedly
• Find a way to guarantee cooperation
• Change payoff structure

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Nuclear arms racesprisoner's dilemma in disguise
Is there a Nash Equilibrium?
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Cigarette Advertisingprisoner's dilemma in
disguise
Philip Morris Philip Morris
No Ad Ad
Reynolds No Ad 50 , 50 20 , 60
Reynolds Ad 60 , 20 30 , 30
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Environmental policyprisoner's dilemma in
disguise
Factory C Factory C
pollution No pollution
Factory R pollution 50 , 50 60 , 20
Factory R No pollution 20 , 60 20 , 20
Two factories producing same chemical can choose
to pollute (lower production cost) or not to
pollute (higher production cost).
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Another Example Big Little Pigs
Cost to press button 2 units
When button is pressed, food given 10 units
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Decisions, decisions...
Whats the best strategy for the little pig? Does
he have a dominant strategy?
Little Pig
Press
Wait
5 , 1
Press
4 , 4
Big Pig
Wait
9 , -1
0 , 0
Does the big pig have a dominant strategy?
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Research in industriesBig Little Pigs in
disguise
Small Company Small Company
research No research
Big Company research 5 , 1 4 , 4
Big Company No research 9 , -1 0 , 0
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Maximin Minimax Equilibrium in a zero-sum game

• Minimax - minimizing the maximum loss
  (loss-ceiling, defensive)
• Maximin - maximizing the minimum gain
  (gain-floor, offensive)
• Minimax Maximin

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Maximin, Minimax Equilibrium Strategies
Opponent
Strategy 2
Strategy 1
Row Min
150
150
1000
Strategy 1
Strategy 2
25
- 10
- 10
You
Strategy 3
-15
-15
160
160
Col. Max
1000
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Saddle point
Row Min
Is this a Nash Equilibrium?
1
3
MaxiMin
4
3
Col. Max
MiniMax
A zero-sum game with a saddle point.
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The Minimax Theorem
Every finite, two-person, zero-sum game has a
rational solution in the form of a pure or mixed
strategy.
John Von Neumann, 1926
For every two-person, zero-sum game with finite
strategies, there exists a value V and a mixed
strategy for each player, such that (a) Given
player 2's strategy, the best payoff possible for
player 1 is V, and (b) Given player 1's strategy,
the best payoff possible for player 2 is -V.
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Two-Person, Zero-Sum Games Summary

• Represent outcomes as payoffs to row player
• Find any dominating equilibrium
• Evaluate row minima and column maxima
• If maximinminimax, players adopt pure strategy
  corresponding to saddle point choices are in
  stable equilibrium -- secrecy not required
• If maximin minimax, find optimal mixed
  strategy secrecy essential

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Summary Look for any equilibrium

• Dominating Equilibrium
• Minimax Equilibrium
• Nash Equilibrium

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Pure mixed strategies
A pure strategy provides a complete definition of
how a player will play a game. It determines the
move a player will make for any situation they
could face. A mixed strategy is an assignment of
a probability to each pure strategy. This allows
for a player to randomly select a pure
strategy. In a pure strategy a player chooses an
action for sure, whereas in a mixed strategy, he
chooses a probability distribution over the set
of actions available to him.
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All you need to know about Probability

• If E is an outcome of action, then P(E) denotes
  the probability that E will occur, with the
  following properties
• 0 ? P(E) ? 1 such that
• If E can never occur, then P(E) 0
• If E is certain to occur, then P(E) 1
• The probabilities of all the possible outcomes
  must sum to 1

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A zero-sum game Matching Pennies
Player 2
Player 1
Maximin minimax no saddle point No pure
Nash Equilibrium for every pure strategy in this
game, one of the players has an incentive to
deviate
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Mixed strategies for matching pennies
Sticking to a single strategy will not lead to
any meaningful solution in matching pennies. So
we try a new type of solution mixing the two
choices together. Assume that player 1 picks
Head with probability p and Tail with
probability 1-p. If player 2 chooses H, he is
expected to gain -p (1-p) 1-2p If player 2
chooses T, he is expected to gain p - (1-p)
-12p If player 1 chooses p such that 1 - 2p
-1 2p, or p1/2, then no matter what player 2
does (choosing H or T) he gets the same
payoff. Similarly, if player 2 mixes H T
together with probabilities 1/2 1/2, then no
matter what player 1 does (choosing H or T) he
gets the same payoff.
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A graphical explanation of Mixed strategies
Payoff for player 2
y(T)2p-1
1
0
1
p probability of choosing H for player 1
-1
y(H)1-2p
Min of the max gain of player 2 max of min loss
of player 1
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Another Mixed strategy example
The game Rock-Paper-Scissors also do not have a
pure strategy equilibrium. In this game, if
Player 1 chooses R, Player 2 should choose p, but
if Player 2 chooses p, Player 1 should choose S.
This continues with Player 2 choosing r in
response to the choice S by Player 1, and so
forth. In games like Rock-Paper-Scissors, a
player will want to randomize over several
actions, e.g. he/she can choose R, P S in equal
probabilities.
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Mixed strategies
xprobability to take action R
yprobability to take action S
x
y
1-x-y
1-x-yprobability to take action P
no
no
No Nash equilibrium for pure strategy
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They have to be equal if expected payoff
independent of action of player 2
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Two-Person, Zero-Sum Game Mixed Strategies
Column Player
Matrix of Payoffs to Row Player
Row Minima
No dominating strategy
A B
0 5 10 -2
1 2
0 -2
Row Player
10 5
Column Maxima
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Two-Person, Zero-Sum Game Mixed Strategies
Column Player
Matrix of Payoffs to Row Player
Row Minima
A B
MaxiMin
0 5 10 -2
1 2
0 -2
Row Player
10 5
Column Maxima
MiniMax
MaxiMin
No Saddle Point!
MiniMax
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Optimized Mixed Strategy Graphical Solution
VR
2A
10
VR lt 0x10(1-x)
1B
VR lt 5x-2(1-x) -2 7x
Optimal Solution x12/17, 1-x5/17 VRMAX50/17
50/17
1A
0
1
x
12/17
Probability of taking action 1
2B
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Graphical Solution
y probability of taking action A
VR
2A
10
VR lt 10(1-x)
y1
y.75
1B
y0
VR lt -2 7x
y.5
Optimal Solution x12/17, 1-x5/17 VRMAX50/17
50/17
y.25
1A
0
1
x
12/17
2B
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x probability taking action 1
1-x probability taking action 2
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Payoffs of player2
1A
2B
Optimal Solution x3/7, 1-x4/7 VRMAX44/7
2A
1B
3/7
x
0
1
Probability of player1 taking action 1
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x probability taking action A
1-x probability taking action B
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Payoff of pure strategy
Payoff of mixed strategy
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Pareto optimal
Nash equilibrium
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N-person games Larger games (More than 2
players) An Example of a 3-person
non-cooperative game Truel
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A truel is like a duel, except that three
players. Each player can either fire, or not
fire, his or her gun at either of the other two
players. The players preferences are lone
survival (the best 4), survival with another
player (the second best 3), all players
survival (the second worst2), the players own
death (worst case1).
If they have to make their choice simultaneously,
what will they do?
Ans. All of them will fire at either one of the
other two players.
If their choices are made sequentially
(AgtBgtCgtAgtBgt) and the game will continue until
only one player survives, what will they do?
Ans. They will never shoot.
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A
dont shoot
shoot C
shoot B
B
B
B
s C
s C
s
s
s C
s A
s A
s
s A
C
s
s B
s A
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Example The paradox of the Chairs Position
Three voters ABC are electing the chairperson
among them. Voter A has 3 votes. Voters B and C
have 2 votes each. Voter As preference is (ABC).
Voter Bs preference is (BCA). Voter Cs
preference is (CAB).

• Who will win if voters vote their first
  preference? (sincere voting)
• Who will win if voters will consider what other
  players may do? (sophisticated voting)

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• If voters vote sincerely,
• Voters A will vote for voter A, voters B will
  vote for voter B, voters C will vote for voter C.
  So, the winner is voter A.

• Lets consider voters A and BC as follows.

A\ (BC) AB BB BC .
A A B A
B B B B
C C B C
So, the dominant strategy for voter A is voting
for A. Assuming voter A will vote for A, lets
consider voters B and C.
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B\C A B C
A A A A
B A B A
C A A C
So, the dominant strategy for voter C is voting
for C. Assuming voters A and C will vote for A
and C respectively, lets consider voter B.
B votes for A B C
result A A C
So, the dominant strategy for voter B is voting
for C. As a result, voters A, B and C will vote
for A, C and C, respectively. So, the winner is
voter C.
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Impact of game theory

• Nash earned the Nobel Prize for economics in 1994
  for his pioneering analysis of equilibria in the
  theory of non-cooperative games
• Nash equilibrium allowed economist Harsanyi to
  explain the way that market prices reflect the
  private information held by market participants
  work for which Harsanyi also earned the Nobel
  Prize for economics in 1994
• Psychologist Kahneman earned the Nobel prize for
  economics in 2002 for his experiments showing
  how human decisions may systematically depart
  from those predicted by standard economic theory

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Fields affected by Game Theory

• Economics and business
• Philosophy and Ethics
• Political and military sciences
• Social science
• Computer science
• Biology