MATH 102Q PROBLEM SOLVING LECTURE 9

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MATH 102Q PROBLEM SOLVINGLECTURE 9

• MISSY PROKOP

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Agenda

• Game Theory
• Introduction
• Definition
• Example
• Assumptions

• Analysis
• Game Matrix Representing the Game
• Matrix Components
• Dominant Strategies
• Nash Equilibrium
• Examples

• Final Group Project Introduction
• Midterm Next Week!
• Problem Set 8 Due Week of 4/10

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Game Theory

• Definition A branch of mathematical analysis
  developed to study decision making in conflict
  situations.
• Background Game Theory was originally developed
  to address problems in economics and has since
  been used in a variety of other fields such as
  political science, psychology, philosophy, and
  biology.
• Objective Game theory provides a mathematical
  process for selecting an optimum strategy (or
  decision) given a specific set of circumstances.

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The Game of Chicken

• Dude and Guy decide to fight it out the macho
  (ehchemstupid) way. They speed down the middle
  of the road towards each other. Each hopes that
  the other will chicken out at the last minute and
  swerve to the right. (Keep in mind that it is
  better to be chickens together than alone.) Is
  there a way to analyze the situation to gain
  insight about their behavior?

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Game Theory Assumptions

• A Game is defined by two players each with two
  possible decisions.
• Players make their decision simultaneously.
• Every possible combination of options leads to a
  well-defined end-state.
• A specified payoff for each player is associated
  with each end-state.
• Each decision maker has perfect knowledge of the
  game and of his opposition that is, he knows in
  full detail the rules of the game as well as the
  payoffs of all other players. He does not know
  the opponents decision.
• All decision makers are rational want to
  optimize profit.

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Game Matrix

• Use the See it method by modeling the game with a
  matrix that gives a representation of the
  possible outcomes.
• Steps
• Create a 3x3 grid

• Fill in the players names

• Label their options

• For each end-state, define the rank/payoff for
  each player.

Guy Dude
Dont forget a key
Best?Worst1?4
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Game Matrix Components
Player 2s Options Cooperate Not
Cooperate
Player 2
Player 1
Best?Worst1?4
Player 1s Options Cooperate Not
Cooperate
This end-state is the combination of dude and guy
not swerving
The End-States All possible outcomes or
combinations
(Player 1 rank, Player 2 rank)
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Analysis Dominant Strategy

• Each player may have a dominant strategy.
• Definition The best decision, considering each
  of the opponents choices
• Procedure To Determine P1s (or P2s) DS
  Isolate each of P2s choices by mentally boxing
  off each column (row). Identify which decision
  will result in the highest payoff by drawing an
  arrow from P1s lower payoff to the higher
  payoff. If the arrows point in the same
  direction, then the optimal decision does not
  depend on P2s (P1s) decision, then the optimal
  decision (to cooperate or not) is the DS.

If Guy swerves, then its better for Dude to not
swerve. However if Guy does not swerve, then its
better for Dude to swerve. Therefore, Dude has no
DS!
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Analysis Nash Equilibrium

• Each game may have between zero and four Nash
  Equilibrium points.
• Definition An end-state such that neither player
  can gain by unilaterally (vertically or
  horizontally) changing strategy.
• Application If a game has one Nash Equilibrium
  point, then it is the rational tendency.
• Procedure To Determine NE Check each end-state
  at a time. For each unilateral change, draw
  arrows in the direction of the higher payoff. If
  both arrows point towards the suspect end-state,
  then you have found a NE. However, if at least
  one arrow is going the wrong way, then the
  end-state is not a NE because that means the
  payoffs will increase with a unilateral change.

One NE is (N,S) when Dude does not swerve and
Guy swerves
Note Use double arrowed lines if the payoffs
stay the same
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Example 1 Elder

• An elderly lady is crossing the street. She
  cannot cross the street alone. Alice and Bob are
  the only two around. Each must decide,
  simultaneously, if he/she will help. Each will
  get a payoff of 3 if the old lady is successful.
  However, considering the time it takes, helping
  her will incur a cost of 1. Model the situation
  with a matrix. Do you suspect she will get help?

Best?Worst3?0
This game has 2 NE and no DS, so this is not
very helpful. However, it is important to Pause
and Reflect on the fact that each player receives
a payoff only if the elderly lady is helped.
Therefore, it seems that she would receive
assistance by both Alice and Bob. (Seems like the
nice thing to do)
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Act 1

• In this game, two people choose (simultaneously)
  whether to show heads or tails of a coin. If they
  show the same side, person 2 pays person 1 a
  dollar. If they show different sides, person 1
  pays person 2 a dollar. Model this game with a
  matrix.

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Act 2 Battle of the Sexes

• A couple wish to go out this Friday night. The
  problem is that they like very different scenes.
  The boyfriends ideal night involves a mud
  wrestling tournament and the girlfriends is a
  romantic candle-lit dinner. They decide to choose
  their destination simultaneously. Be aware that
  the worst possible case would result in a
  disagreement which would lead to an evening spent
  sulking at home. Find any dominant strategies and
  Nash Equilibrium points.