Ch 10

• Game theory


  A general framework to aid decision making when agents' payoffs depends on the actions taken by other players
• Components of games


  • Players or agents who make decisions
  • Planned actions of players, called strategies
  • Payoff of players under different strategy scenarios
  • A description of the order of play
  • A description of the frequency of play or interaction
• Simultaneous-move game


  Game in which each player makes decisions without the knowledge of the other players' decisions
• Sequential-move game


  Game in which one player makes a move after observing the other player's move
• One-shot game


  Game in which players interact to make decisions only once
• Repeated game


  Game in which players interact to make decisions more than once
• Strategy


  Decision rule that describes the actions a player will take at each decision point
• Normal-form game


  A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies
• Components of a normal form game

  • Set of players
  • Each player's strategy set or feasible actions
  • Payoffs
• Dominant strategy


  A strategy that results in the highest payoff to a player regardless of the opponent's action
• Secure strategy


  A strategy that guarantees the highest payoff given the worst possible scenario
• Nash equilibrium strategy


  A condition describing a set of strategies in which no player can improve his payoff by unilaterally changing his own strategy, given the other players' strategies
• Scenario analysis in a normal form game


  1. Suppose 1 thinks 2 will choose "A", then 1 should choose "a"

  2. Suppose 1 thinks 2 will choose "B", then 1 should choose "a"

  3. Similarly, if 1 thinks 2 will choose "C", then 1's best response is "a"
• Dominant strategy


  Regardless of whether Player 2 chooses A, B, or C, Player 1 is better off choosing "a"
• Player 1 has a dominant strategy


  Player 1 should choose "a"
• Player 2 has no dominant strategy


  Player 2 should choose "C" since Player 1 will choose "a"
• Nash equilibrium


  The outcome where "a" is player 1's best response to "C", and "C" is player 2's best response to "a"
• The Nash equilibrium is a condition describing the set of strategies in which no player can improve his payoff by unilaterally changing his own strategy, given the other player's strategy
• Nash equilibrium


  • May come from dominant strategies, but doesn't require them
• Secure strategy


  Player A's secure strategy: Up ... guarantees at least a $10 payoff

  Player B's secure strategy: Right ... guarantees at least an $8 payoff
• The Nash Equilibrium, however, is "up" and "left"
• Application: A Pricing Game

  Two managers want to maximize profit

  Strategies are pricing decisions: S1 = {low, high}, S2 = {low, high}

  Simultaneous moves, one-shot game

  A Nash equilibrium results when both players charge "Low price"

  Both firms have dominant strategy of low price

  Payoffs associated with the Nash equilibrium are inferior from the firms' viewpoints compared to both "agreeing" to charge "High price": hence, a dilemma

  An incentive to collude exists (but with temptation to cheat)
• Application: An Advertising Game

  Two managers want to maximize market share

  Strategies are pricing decisions: S1 = {1, 5, 10}, S2 = {1, 5, 10}

  Simultaneous moves, one-shot game

  Nash Equilibrium
• Coordination games


  Games where players have competing objectives, but some games result in higher profits by each firm when they "coordinate" decisions
• Examples of coordination games


  • Industry standards (size of floppy disks, size of CDs, usb standards)

  National standards (electric current, traffic laws)
• A coordination game in normal form

  Two Nash equilibrium outcomes: 1) Both players choose 120-volt outlets, 2) Both players choose 90-volt outlets

  Ways to coordinate on one equilibrium: 1) permit player communication, 2) government set standard
• Nash bargaining


  Management and a union are negotiating a wage increase

  Strategies are wage offers & wage demands

  Successful negotiations lead to $600 million in surplus, which must be split among the parties

  Failure to reach an agreement results in a loss to the firm of $100 million and a union loss of $3 million

  Simultaneous moves, and time permits only one-shot at making a deal
• Nash equilibria


  Three Nash equilibria in the bargaining game
• Mixed strategies


  In games where no pure strategy Nash equilibrium exists, players find it in their interest to engage in mixed (randomized) strategies
• Considering repeated games: An advertising game

  Two firms' (Kellogg's & General Mills) managers want to maximize profits

  Strategies consist of advertising campaigns

  Simultaneous moves, one-shot interaction, repeated interaction
• Equilibrium in the one-shot advertising game

  No dominant strategy, Nash equilibrium
• Collusion cannot work if the game is repeated a known, finite number of times (by "backwards induction")
• Infinitely repeated game


  A game that is played over and over again forever, and in which players receive payoffs during each play of the game
• One-shot interaction


  A single interaction between players with no future interactions
• Repeated interaction


  An interaction between players that occurs multiple times
• Simultaneous moves


  Players make decisions at the same time without knowing the other's choice
• Advertising campaign


  A series of advertising activities designed to promote a product or service
• One-Shot Advertising Game


  • General Mills
  • Kellogg's
• Nash Equilibrium


  A set of strategies where no player can improve their payoff by unilaterally changing their strategy
• In the one-shot advertising game, there is a Nash Equilibrium but no dominant strategy