Game Theory and Imperfect Competition

Game Theory and Imperfect Competition

Introduction to Game Theory

• Game theory is a framework for modeling scenarios in which players make decisions that are interdependent.

• The outcome for each participant depends not just on their own decision but also on the decisions made by others.

Key Concepts

Normal Form Games

• Definition: Normal Form Games

  • A setting with $I$ players (firms, individuals, countries) where each player $i$ chooses a strategy $si$ from a strategy set $Si$

  • Strategy profile $(si, s{-i})$ denotes the strategies selected by all players except player $i$, where $s{-i} = (s1,\, …, s{i-1}, s{i+1}, …, s_I)$

  • Payoffs can represent different factors such as output levels, prices, or advertising expenditures.

Dominated Strategies

• Dominated Strategy: A strategy $si^$ strictly dominates another strategy $si' \neq si^$ for player $i$ if \pii(si^*, s{-i}) > \pii(si', s{-i}) for all $s{-i}$

  • This implies $si^*$ yields a strictly higher payoff than $si'$ regardless of the strategies selected by other players.

Example of Dominated Strategies:

• 
  

• Payoff Matrix:
  

  	Firm B Low Prices	Firm B High Prices
  Firm A Low Prices	5, 5	9, 1
  Firm A High Prices	1, 9	7, 7
  "Low prices" is strictly dominant for both firms
  Both firms will select "low prices" in equilibrium.
  Nash Equilibrium (NE)

  • Nash Equilibrium (NE): A strategy profile $(si^, s{-i}^)$ is a NE if for every player $i$, $\pi<em>i(s</em>i^<em>, s{-i}^) \geq \pi</em>i(s<em>i, s</em>{-i}^<em>) \text{ for all } si \neq si^</em>$

    • Meaning, $s_i^*$ is player $i$’s best response to the strategies chosen by others.

  Mixed-Strategy Nash Equilibrium (msNE)

  • Definition:

    • A strategy profile $\sigma = (\sigma1, \sigma2, …, \sigman)$, where $\sigmai$ is a mixed strategy for player $i$.

    • It is a msNE if
      $\pi<em>i(\sigma</em>i, \sigma<em>{-i}) \geq \pi</em>i(s<em>i', \sigma</em>{-i})$ for all $si' \in Si$$

  • Players are indifferent among their pure strategies, and dominated strategies get assigned a zero probability.

  • In finite games, there usually exists an odd number of equilibria.

  Example of msNE:

  • The probability of firm $A$ adopting technology is denoted $q$. If indifferent between adopting and not adopting, the expected utility is calculated.

  • Derived probabilities, such as $q = 1/2$, indicate the msNE results.

  Sequential Move Games

  • In games where players choose strategies sequentially rather than simultaneously, the definition of strategy becomes more complex.

  • Strategy: A complete contingent plan for what action player $i$ chooses given the history of play.

  • Represented using game trees with an initial node (where the game starts) and terminal nodes (where no more branches arise).

  Backward Induction

  • To find equilibria in sequential games, players utilize backward induction by starting from terminal nodes, moving backward to identify optimal choices at every decision point or information set.

  • Subgame Perfect Nash Equilibrium (SPNE): A strategy profile is a SPNE if it specifies a NE for each subgame.

  Simultaneous-Move Games of Incomplete Information

  • Incomplete Information: Occurs in scenarios where at least one player cannot observe a piece of information about the game. Players are referred to by their "type", denoted as $\theta_i$.

  • Bayesian Nash Equilibrium (BNE): A strategy profile $(s1^* \theta1, s2^* \theta2, …, sN^* \thetaN)$ is a BNE if the expected utility calculations hold under the assessment of types.

  Conclusion

  • Understanding the nuances of complete vs. incomplete information and how strategies adjust based on players’ knowledge ensures a robust analysis of competitive behavior in game theory.

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