SCIT1003 Chapter 1: Introduction to Game theory

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SCIT1003Chapter 1 Introduction to Game theory

• Prof. 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 ???
Chess Combinatorial games
Gambling Games of pure luck
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Game Theory ???

• Game theory is a study of strategic decision
  making.
• Specifically "the study of mathematical models
  of conflict and cooperation between intelligent
  rational decision-makers".
• Game theory is mainly used in economics,
  political science, and psychology, as well as
  logic, computer science, and biology.
• 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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What does game mean? according to Webster

• 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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In a nutshell Game theory is

• the 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

• Studies of military strategies dated back to
  thousands of years ago (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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???? ???? ???? Putting yourself in the other
persons shoes
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Rationality

• Assumptions
• Humans are rational beings
• They are in the game to win (get rewarded)
• 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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Whats good for you?Utility Theory

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

Utility (usefulness) is an economic concept,
quantifying a personal preference with respect to
certain result/reward as oppose to other
alternatives. It represents the degree of
satisfaction experienced by the player in
choosing an action.
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Utility Example (Exercise)

• Which would you choose? (Game is only played
  once!)
• 10 million Yuan (100 chance), or
• 100 million Yuan (10 chance)
• Which would you choose? 10 Yuan (100 chance),
  or 100 Yuan (10 chance)

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What are the Games in Game Theory?

• In Game Theory, our focus is on games where
• There are 2 or more players.
• Where strategy determines players choice of
  action.
• 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.
• Examples
• Chess, Go, economic markets, politics, elections,
  family relationships, etc.

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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. repeated game

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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 (strategies).
• The information that players have available when
  choosing their actions
• A description of the payoff (reward) consequences
  for each player for every possible combination of
  actions chosen by all players playing the game.

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Characteristics of Game Theory

• Fundamentally about the study of decision-making
• Investigations are concerned with choices and
  strategies of actions available to players.
• It seeks to answer the questions
• What strategies are there?
• What kinds of solutions are there?
• A solution is expressed as a set of strategies
  for all players that yields a particular payoff,
  generally the optimal payoff for all players.
  This payoff is called the value of the game.

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Games economics

• Games are convenient ways to model strategic
  interactions among economic agents.
• Many economic situations involve strategic
  interactions
• Behavior in competitive market e.g. Coca Cola
  vs. Pepsi
• Behavior in auctions bidders bidding against
  other bidders
• Behavior in negotiations e.g. trade negotiations

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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 (rewards).

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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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Zero-Sum none zero-sum Games

• When the interests of both sides are in conflict
  (e.g. chess, sports) the sum of the payoffs
  remains zero during the course of the game.
• A game is non-zero-sum, if players interests are
  not always in direct conflict, so that there are
  opportunities for both to gain.

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As a rational game-player, you should

• Know the payoffs of your actions.
• Know you opponents payoff.
• Choose the action that maximizes your payoff.
• Expect your opponents will do the same thing.
• Putting yourself in the other persons shoes

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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

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Game Theory in the Real World

• Economists
• innovated antitrust policy
• auctions of radio spectrum licenses for cell
  phone
• trade negotiation.
• Computer scientists
• new software algorithms and routing protocols
• Game AI
• Military strategists
• nuclear policy and notions of strategic
  deterrence.
• Politics
• Voting, parliamentary maneuver.
• Biologists
• How species adopt different strategies to
  survive,
• what species have the greatest likelihood of
  extinction.

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Summary Ch. 1

• Essentials of a game
• Payoffs (Utilities)
• Normal Form Representation (payoff matrix)
• Extensive Form Representation (game tree)

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Assignment 1.1
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Assignment 1.2
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Assignment 1.3
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Assignment 1.4 draw the game tree for the game
Simple Nim

• (Also called the subtraction game)
• Rules
• Two players take turns removing objects from a
  single heap or pile of objects.
• On each turn, a player must remove exactly one or
  two objects.
• The winner is the one who takes the last object
• Demonstration http//education.jlab.org/nim/index
  .html

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Assignment 1.5 Hong Kong Democratic Reform game
Demo-parties
Accept Reject
No reform ? ? ? ?
Gradual reform ? ? ? ?
One-step reform ? ? ? ?
Central Government
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Assignment 1.5 Hong Kong Democratic Reform game
The democratic reform process in Hong Kong can be
regarded as a 2-player game. On one side is the
Central Government in Beijing. On the other side
is the democratic parties in the Legislature
Council in Hong Kong. According to the Basic Law
of the Hong Kong SAR (Special Administrative
Region), the Central Government proposes the law
for the democratic reform and the democratic
parties in Legislature Council can either pass
or reject the law. Reform can move forward only
if the Central Government proposes the law and
the democratic parties in the Legislature Council
accept and pass the law. The Central Government
can propose law that contains no reform, gradual
reform, or one-step (radical) reform, and the
democratic parties can accept or reject the law.
a Assuming the Central Government prefers
gradual reform to no reform to radical reform,
and the democratic parties prefers radical reform
to gradual reform to no reform, choose and
justify some simple numerical payoffs for this
game in normal form. b Is this a zero
(constant) sum or non- zero (constant) sum
game? c Is this a cooperative or
non-cooperative game? d Is this a complete or
incomplete information game? e Is there a
solution to this game if all players are
rational? Explain your answers.