Oligopoly games & strategies: Prisoners dilemma

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Intros
Lessons
  1. Oligopoly Games & Strategies: Prisoner's Dilemma
  2. Definition of a Game
    • Rules
    • Strategies
    • Payoffs
    • Outcome
    • We will see these in Prisoner's Dilemma
  3. Rules of Prisoner's Dilemma, and Strategies
    • Background of Prisoner's Dilemma
    • Rules of each prisoner
    • Strategies of each player
    • Possible Outcomes
  4. The Payoff Matrix & Nash Equilibrium
    • What each player gains/loss in each outcome
    • Predicting the Outcome
    • Nash equilibrium
    • Best choice given the action of prisoner B
    • Best choice given the action of prisoner A
    • Equilibrium of the Game: Worst Outcome
  5. Prisoner's Dilemma in Duopoly
    • Collusive Agreement (Comply or Cheat)
    • Possible Outcomes for Complying or Cheating
    • What happens to each firm in each outcome
    • Payoff Matrix & Nash Equilibrium of Duopoly
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Examples
Lessons
  1. Understanding Game Definitions
    Determine whether each action is allowed in a prisoner's dilemma.
    1. Accuse each other
    2. Stay silent
    3. Communicate with each other
    4. Negotiate with the police
    5. Accuse first, then stay silent
    6. Stay silent first, and then accuse.
  2. Determine whether each statement is true or false in a prisoner's dilemma.
    1. There are always 4 outcomes
    2. Players can have more than two strategies
    3. The players cannot communicate with each other.
    4. Players always make choices base on their personal gain.
  3. Calculating the Payoff Matrix & Nash Equilibrium in Prisoner's Dilemma
    Suppose there is a two-player game in which they cannot communicate. Each player is asked a question and must answer honestly or lie. If both answer honestly, then each receives $100. If one answers honestly and one lies, then the liar receives $500 and the honest player gets nothing. If both lie, then each player receives $50.
    1. Describe the strategies and outcomes
    2. Make the payoff matrix.
    3. What is the Nash equilibrium?
  4. Suppose there are two prisoners in which they cannot communicate. Each prisoner is interrogated by the police. If both prisoners stay silent, then both will be sentenced to jail for 2 years. If both accuse each other, then both will be sentenced to jail for 4 years. If one accuses and the other stays silent, then one of them will be sentenced to jail for 7 years, and the other prisoner will be free.
    1. Describe the strategies and outcomes.
    2. Make the payoff matrix.
    3. What is the Nash equilibrium?
  5. Calculating the Payoff Matrix & Nash Equilibrium with Duopoly
    Firm A and firm B are both producers of soft drinks. Both firms are trying to figure out how much soft drinks is needed to be produced. They know the following:
    1. If both limit productions to 10,000 gallons a week, they will make a maximum attainable profit of $100,000. So, each firm gains $50,000 a week.
    2. If one firm produces 10,000 gallons a week and the other firm produces 5,000 gallons a week, then the one who produces 10,000 gallons will gain an economic profit of $100,000, while the other will incur an economic loss of $25,000.
    3. If both increases their production 10,000 gallons a week, both firms will gain zero economic profit.

    1. Construct a payoff matrix.
    2. Find the Nash equilibrium.
  6. Firm A and firm B are both producers of wallets. The firms decide to collude and form a cartel to share the market equally. If neither firm cheat on the agreement, then each makes $2 million profit. If one firm cheats, then the cheater make a profit of $3 million and the complier will lose $1 million. If both cheats, they will both break even. Assume that none of the firms can monitor the other actions.
    1. What are the strategies of the game?
    2. Construct the payoff matrix.
    3. What is the Nash equilibrium?
Topic Notes
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Definition of a Game

In an economic sense, firms in an oligopoly market play a game with each other to try to achieve economic profit.

Lets first try to investigate the features of a game, and then later see how it relates to Prisoners Dilemma and Duopoly.

There four common features in a game:
  1. Rules: laws that need to be followed.
  2. Strategies: possible actions for a player.
  3. Payoffs: the gains and loss of a player from his actions.
  4. Outcomes: the result gained from the action of all players.

Rules of Prisoners Dilemma, and Strategies

Background: Suppose two people are suspects of a crime, and now must become prisoners. We will call them prisoners A and B. The police do not know who committed the crime, so they force the prisoners to play a game to see if they stay silent or betray each other.

Note: Assume that the prisoners are not friends, and do not know each other.

Rule: Each prisoner is placed in an isolated room, and the prisoners cannot communicate with each other. Each prisoner is told that they are suspected of committing the crime, and are told the following
  1. If both do not accuse each other of committing the crime and stay silent, then both will be sentenced to jail for 3 years.
  2. If the prisoner accuses the other prisoner of committing the crime and the other stays silent, then the accuser is free, and the other prisoner is sentenced to jail for 10 years.
  3. If both prisoners accuse each other, then both will be sentenced to jail for 5 years.

Strategies: Prisoner A and B have two possible actions
  1. Stay silent to the crime.
  2. Accuse the other prisoner.

Outcomes: Since there are two prisoners and each have two strategies
  1. Both stay silent
  2. Prisoner A accuses, Prisoner B stays silent
  3. Prisoner A stays silent, Prisoner B accuses
  4. Both prisoners accuse each other

The Payoff Matrix, & Nash Equilibrium

A payoff matrix is a table that shows all the strategies each player can make, and lists the gains/losses of each player for every outcome.
The Payoff Matrix, & Nash Equilibrium


Nash Equilibrium: is the equilibrium where each players strategy is optimal when given the strategies of all other players.

Lets look at the optimal strategy for prisoner A.
  • Case 1: Prisoner A assumes Prisoner B stays silent about the crime.

    If prisoner A chooses.

    1. \enspace Stay silent  \, →  \, 3 years in jail
    2. \enspace Accuses  \, →  \, 0 years in jail

    The best choice is to accuse prisoner B.
    • Case 2: Prisoner A assumes Prisoner B accuses.

  • If prisoner A chooses.

    1. \enspace Stay silent  \, →  \, 10 years in jail
    2. \enspace Accuses  \, →  \, 5 years in jail

    The best choice is to accuse prisoner B.


So, whether prisoner B stays silent or accuses, prisoner As best action is to accuse.

Lets look at the optimal strategy for prisoner B.
  • Case 1: Prisoner B assumes Prisoner A stays silent about the crime.

    If prisoner B chooses.

    1. \enspace Stay silent  \, →  \, 3 years in jail
    2. \enspace Accuses  \, →  \, 0 years in jail

    The best choice is to accuse prisoner A.
    • Case 2: Prisoner B assumes Prisoner A accuses.

  • If prisoner B chooses.

    1. \enspace Stay silent  \, →  \, 10 years in jail
    2. \enspace Accuses  \, →  \, 5 years in jail

    The best choice is to accuse prisoner A.


So, whether prisoner A stays silent or accuses, prisoner Bs best action is to accuse.

Hence, the Nash equilibrium is for both prisoners accuse each other. This outcome will lead both prisoners to go to jail for 5 years.

Prisoners Dilemma in Duopoly

The same idea in prisoners dilemma holds for duopoly.

Collusive Agreement: an agreement between two firms to form a cartel and act as a monopoly.

Suppose there is firm A and firm B, and the demand and costs for the product is below.

Collusive Agreement

Strategies: Each firm can either
  1. Comply
  2. Cheat .

Outcomes:
  1. Both firms Comply: then they enter a collusive agreement and act as a monopoly. Both firms produce an output of 15, and sell each outprice for $5.

    2. Both firms Comply Collusive Agreement act as a monopoly

    Result: Both firms gain an economic profit of $7.50

  2. Firm A Complies, and Firm B Cheats: Then firm B produces 10 more than firm A, thus lowering the price to $4.
    3. Collusive Agreement Firm A Complies, and Firm B Cheats

    Result: Firm B gains an economic profit of $37.50, and Firm A incurs an economic loss of -$7.50.

    Collusive Agreement Firm A Complies, and Firm B Cheats

  3. Firm B Complies, and Firm A Cheats: Vice versa happens in outcome b.

  4. Both Firms Cheat: They break the collusive agreement and both firms will increase their output to 25, thus lowering the price to $2.50. Both firms will gain no profit.
Collusive Agreement Both Firms Cheat

Payoff: With the listed outcomes, we can create a payoff matrix.
Collusive Agreement payoff matrix outcomes

The Nash equilibrium will be for both firms to cheat.