Intermediate Microeconomics Game Theory and Oligopoly
Game Theory • So far we have only studied situations there were not “strategic”. • The optimal behavior of any given individual or firm did not depend on what other individuals or firms did. • E.g. • An individual buys something if its price is less than his willingness to pay. • A firm enters a market if there are positive economic profits to be made.
Game Theory • Game theory helps to model strategic behavior --- or interactions where what is optimal for a given agent depends on what actions are taken by another agent and vice versa. • Applications: • The study of oligopolies (industries containing only a few firms) • The study of externalities and public goods; e.g. using a common resource such as a fishery. • The study of military strategies. • Bargaining. • How markets work. • Behavior in the courts.
What is a Game? • A game consists of: • a set of players • a set of strategies for each player (i.e. actions to be performed given any observed state of the world) • the payoffs to each player for every possible choice of actions by the players.
Simultaneous Move Games • Consider games where players must choose an action without knowing what the other players have chosen. • Does a defendant agree to cooperate with prosecutors against his co-defendants when he doesn’t know whether or not his co-defendants have “cracked”? • How much should a firm bid for a given item in a silent auction? • Should I act friendly or defensively when I encounter a stranger on an empty street late at night? • How do we model the outcomes in these types of games?
Simultaneous Move Games • Consider a game with 2 players, each player has two options: • Keep quiet (Q) or Cooperate with Police (C). • Payoffs to Player-1; • Quiet utility of 5 if Player-2 Quiet, utility of 2 if Player-2 Cooperates. • Cooperate utility of 8 if Player-2 Keeps Quiet, utility of 3 if Player-2 Cooperates. • Payoffs to Player-2 are analogous. • “Prisoners’ Dilemma”
One way to summarize the payoffs associated with each action is to use a payoff matrix. Payoff for Player-1 (row player) shown first, followed by payoff for Player-2 (column player) Simultaneous Move Games Player-2 Q C Q C Player-1
Simultaneous Move Games • So how should we think of how to model the outcome of such games? • What is Player-1’s best action to take if Player-2 Keeps Quiet? • What is Player-1’s best action if Player-2 Cooperates with Police? • How about for Player-2? • So what do think each Player will do? Player-2 Q C Q C Player-1
Simultaneous Move Games • Dominant Strategy - A strategy that gives higher utility than all other strategies given any actions taken by other players. • Does a dominant strategy always exist? • Meeting time for dinner? • Wearing a costume to a Halloween party? • Attacking Iraq?
Simultaneous Move Games • Consider a game of the following form: • What is Player-1’s best action to take if Player-2 chooses Knife? • What is Player-1’s best action to take if Player-2 chooses Gun? • How about for Player-2? • “Arms Race” Player-2 Knife Gun Knife Gun Player-1
Nash Equilibrium • Nash Equilibrium – A set of actions such that each person’s action is (privately) optimal given the actions of others. • Key to a Nash Equilibrium: • No person has an incentive to deviate from his Nash equilibrium action given everyone else behaves according to their Nash equilibrium action. • Nash Equilibrium in “A Beautiful Mind?”
Simultaneous Move Games • Nash Equilibria of “Prisoner’s Dilemma”? • Both Keep Quiet? • One Keep Quiet, other cooperate with Police? • Both cooperate with Police? Player-2 Q C Q C Player-1
Simultaneous Move Games • Nash Equilibria of Arms Race? • Both bring Knife? • One bring Knife, other Gun? • Both Defect? Player-2 K G K G Player-1
Nash Equilibria • Two things to notice: • Nash Equilibria do not have to be Pareto Efficient. • There can be multiple equilibria (often that can be Pareto ranked). • Applications of these types of games?
Game Theory Application: Trade • Suppose Acme Corp. could make a deal with China Corp. to produce widgets abroad. • If both stick with the deal (i.e. China Corp. produces quality widgets and Acme Corp. pays China Corp. the agreed upon fee), Acme’s profits will be $200K while China Corp’s profits will be $50K. • If Acme cheats and pays less than the agreed upon rate after delivery, Acme has profits of $250K and China Corp. ends up losing $50K. • Alternatively, if Acme acts honestly, but China Corp. cheats and produces sub-standard widgets, Acme Corp.’s profits will only be $50K, but China Corp.’s profits will be $90K. • If both act dishonestly, Acme will make only $75K while China Corp. will lose $20K. • If Acme produces widgets domestically, its profits will only be $100K and China Corp. will have profits of $0. • Should trade happen? Will trade happen?
Game Theory Application: Corruption • Solution to game sometimes may depend on characteristics of those playing it. • Consider following model of police corruption. • Suppose cops’ utility function is given by: • Uc = m0.5 – 1, if he chooses to accept a bribe (m is the amount of money he ends up with). • Uc = m0.5 + 1, if he chooses to prosecute a briber • Uc = m0.5, if he doesn’t get a bribe offer. • Suppose speeder can get out of $175 ticket if cop accepts his $50 bribe. Further, let his utility be as follows: • If he offers a bribe of $50 and it is accepted, his utility is Us = (m- 50)0.5 • If he offers a bribe of $50 and is prosecuted, his utility is Us = -100 • If he doesn’t offer a bribe, his utility is Us = m0.5 . • Suppose speeder has $400 at the time he is pulled over. • How will solution differ if cop is poorly paid (i.e. earns $100 without bribes), versus if the cop is well paid (i.e. earns $400 without bribes)?
Continuous Actions • What if players could choose among a continuum of actions. • Ex. Speeder could offer any amount to bribe cop, and cop could decide to accept some bribe amounts but not others. • What will be Nash Equilibrium strategy for cops and speeders given High-paid cops? • How about given Low-paid cops?
Continuous Actions • To handle such situations, we use Reaction Functions. • Reaction function – a function that maps any possible action by Player b into optimal action for player a. • A Nash Equilibrium will arise at the point where Reaction functions intersect. • Why?
Continuous Actions: Cournot Equilibrium • So far, we have examined 2 types of market structure. • Markets where each supplier was small enough that its decision regarding how much to supply had no effect on price (competition) • Markets where there was only one supplier, so its decision regarding how much to supply fully determined price (Monopoly) • What happens with “a few” suppliers?
Continuous Actions: Cournot Equilibrium • Consider a market with two firms, each with cost function equal to C(q) = 4q + 10 • Suppose the market (inverse) demand function is p(Q) = 84 – 8(Q) where Q = q1 + q2 • How do we find optimal quantity supplied by each firm?
So each firm’s reaction function will be: So what will be Nash Equilibrium? Will it be efficient? Continuous Actions: Cournot Equilibrium q1 10 Firm 2’s reaction fn 5 Firm 1’s reaction fn q2 5 10
Continuous Actions: Cournot Equilibrium • What happens as the number of firms increases? • Firm 1 : max p(Q)q1 – C(q1) • FOC: • So what happens when s1 = 1? How about when then number of firms becomes larger, so s1 gets smaller? What about when s1 goes to zero?
