Lecture 4

• Normal form


  Makes use of a matrix to show the pays offs to each 'player' from choosing a particular action (strategy)
• Extensive form


  Uses 'game trees', which depict the sequence of strategies, the decisions for each 'part' of the game, and the game in total in total. All pay-offs are known at all points in the game.
• Pure strategy


  No randomization; player chooses one of the available strategies, and sticks to it
• Mixed strategy


  Strategic games in which the player chooses according to a probability distribution; their choices are not 'fixed'
• In the normal form, each firm makes its decision without knowledge of the other firm's decision and is most useful when players move simultaneously
• In the normal form, we assume the game is only played 'once' (though this can be relaxed)
• A has a dominant strategy in the normal form matrix
• B does not have a dominant strategy in the normal form matrix
• Nash equilibrium


  No player can improve their payoff by unilaterally changing their strategy

  Example: ‘up’ and ‘left’ of Nash equilibrium
• Some games may not have a Nash equilibrium, while other games may have more than one Nash equilibrium
• A Nash equilibrium need not be Pareto optimal
• Matching pennies


  1. Two players secretly choose heads or tails

  2. They reveal their choices simultaneously

  3. If their choices match (two evens , or two odds), player 1 receives a penny from player 2

  4. The opposite applies if their choices do not match (one even, and one odd)
• In matching pennies, there is no Nash equilibrium
• In a mixed strategy, A might choose to play top 50% of the time, and bottom, 50%; and B might choose to play left 50% of the time, and right, 50%
• In a mixed strategy, the players have a probability of 0.25 of ending up in each of the 4 cells of the payoff matrix
• Nash equilibrium in mixed strategies

  Each player chooses his best response probability to his opponent's best response probability
• Prisoner's dilemma


  • Payoffs in each matrix are all negative - time in jail has a negative impact on utility
  • Confess
  • Deny
  • Decision is made independently
• In the prisoner's dilemma, confess/confess is a Nash equilibrium and a dominant strategy equilibrium
• In the prisoner's dilemma, confess/confess is not Pareto efficient
• Repeated games


  • Allows players to trust each other as the game is played many times
  • Allows players to punish each other in subsequent games ('tit-for-tat')
  • Much depends on whether the game is played a finite number of times or indefinitely
• In a finite repeated game, the Nash equilibrium is the same as in a one-shot game
• Infinitely repeated games


  • Game is played forever, and players receive a payment in each round
  • Each player has opportunity to influence his opponent’s behaviour
  • If player A does not cooperate this game, player B refuses to cooperate next game: tit-for-tat
  • Provided both players care enought about future payoffs, the threat of non-cooperation in the future may be sufficient to convince each player to play the Pareto efficient strategy
• Present value


  The value today of a future stream of payments, discounted at an appropriate interest rate
• The critical interest rate is the rate at which the present value of cheating equals the present value of future payments from not cheating
• Tit for tat


  Player A cheats in round 1, so Player B cheats in round 2
• Grim trigger strategy


  If player A cheats in round 1, player B will always cheat in the future
• If the present value of the one-time gain from cheating is less than the present value of what is lost by cheating, both players will abide by the agreement
• Extensive form (multi-stage) games


  • Represents a game in the form of a tree diagram showing the players, available sequences of moves, and their associated pay-offs
  • The first player to move cannot base their decisions on how the second player might react, but the second player can base their decision on what decisions are made by the first player
  • All payoffs are known to each player at the start of the game
  • Backward induction ensures that optimal decision making is embedded in the analysis
• Subgame perfect (Nash) equilibrium


  Equilibrium strategies are Nash equilibria in each subgame
• In the extensive form game, A will play 'up', and B will play 'up' because 10 > 5 for A and 15 > 5 for B
• In the extensive form game, B's threat to play 'down' is not credible if A chooses 'up', since 5 < 15
• Look at lecture 4 for examples


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