10.4

• Game theory is the study of how individuals or firms make decisions when they are interdependent—that is, when the outcome for one player depends not only on their own decisions but also on the decisions made by others.
• the key concepts in game theory:

  • game
  • strategy
  • dominant strategy
  • nash equilibrium
• Game: A situation in which decisions are interdependent, and players (firms) choose strategies to maximize their own payoffs.
• • Strategy: A complete plan of action a player will follow given any circumstances. A firm’s strategy may involve setting output levels, prices, or any other action that impacts profits.
• Dominant Strategy: A strategy that yields the best outcome for a player regardless of what the other player does. If a player has a dominant strategy, they will always follow it because it guarantees the best result.
• Nash Equilibrium: A situation in which each player has selected a strategy, and no player can improve their payoff by changing their strategy while the other players’ strategies remain the same. In other words, no player has an incentive to deviate from their chosen strategy.
• The Prisoner’s Dilemma is a classic example in game theory, showing why two rational individuals might not cooperate, even if it is in their best interest to do so. In an oligopoly, it represents the tension between collusion (acting together) and competition.
• Nash Equilibrium: The Nash equilibrium occurs when both firms choose high output (top-left cell). Each firm chooses high output because it is their dominant strategy. Firm A will choose high output because, regardless of Firm B’s choice, high output gives Firm A a better payoff. The same applies to Firm B
• The nash equilibrium is a dilemma because:

  • Even though both firms would be better off if they cooperated (choosing low output, earning 2 each), they end up choosing high output and earning just 1 each.
  • This occurs because each firm is trying to maximize its own profit without knowing what the other will do, and without an enforceable agreement to collude, they are both worse off than they could have been.
• In a one-shot game like the Prisoner's Dilemma, it is hard for firms to cooperate because they each have an incentive to cheat. However, in repeated games (where firms interact over and over), they can build trust and use commitments to ensure cooperation.
• Commitment: If both firms could sign a binding contract to limit output, they would end up in the cooperative outcome (bottom-right cell), earning 2 each. Without this binding commitment, each firm is tempted to cheat, going for high output to increase their short-term profits.
• Repeated Games: In the real world, firms interact repeatedly. Firms can use punishment strategies (like tit-for-tat) to discourage cheating. For example, if one firm cheats by increasing its output, the other firm may retaliate by increasing its output in future periods, reducing profits for both.
• Credible Threats: A credible threat is one that is believable and enforceable. In the case of repeated games, if one firm cheats (by producing high output), the other firm may credibly threaten to retaliate by producing high output in future rounds. This punishment can prevent cheating and sustain cooperation.
• 1. Repeated Games and Collusion: In industries like oil (e.g., OPEC), firms have long-term relationships. Even if there’s no formal agreement, firms may cooperate by setting low output levels, knowing that cheating will lead to retaliation and lower profits for everyone in the long run.
• A dominant strategy is the best strategy for a player, no matter what the other player does.
• A Nash equilibrium occurs when no player has an incentive to change their strategy, given the strategy chosen by the other player.
• In real-world oligopolies, credible threats of punishment help prevent cheating and can sustain collusion over time.
• • Firms in oligopolies are interdependent, so they must consider how their competitors will respond to their pricing and output decisions. Game theory provides a framework for analyzing these strategic decisions.
• The Nash equilibrium is a point where neither player (firm) can improve their payoff by changing their strategy, assuming the other firm sticks with its strategy.
• both firms choosing high output is the Nash equilibrium, because:

  • If Firm A knows that Firm B will choose high output, Firm A also prefers to choose high output (profit of 1 is better than 0).
  • Similarly, Firm B will choose high output if it believes Firm A will choose high output, for the same reason.
  • Even though this is not the best possible outcome for both firms (since both would earn more if they cooperated), neither firm has an incentive to deviate from high output in this situation, leading to a stable equilibrium.
• • Prisoner's Dilemma:
  • Shows why firms in an oligopoly may fail to cooperate even when it is in their best interest to do so.
  • Dominant strategy for both firms is to choose high output, leading to a Nash equilibrium with lower profits.
  • Cooperation (choosing low output) would maximize joint profits, but the temptation to cheat makes it difficult.
• Nash Equilibrium:

  • Occurs when neither player has an incentive to change their strategy given the strategy of the other player.
  • In the Prisoner's Dilemma, both firms choosing high output is the Nash equilibrium, even though they would be better off cooperating.
• Reaction Functions:

  • Shows how a firm's output decisions depend on the output decisions of its competitor in a duopoly.
  • As one firm increases its output, the other firm’s optimal output decreases because it faces a smaller residual demand curve.
• Commitment and Repeated Games:

  • Firms can sustain cooperation through credible threats and commitments in repeated games.
  • Without an enforceable agreement, firms may revert to the non-cooperative Nash equilibrium.