Introduction to Game Theory

What is Game Theory?

• Studies settings where one individual's actions affect others' payoffs.

• Examines interactions between a group of rational agents who behave strategically.

• ### Key Elements of the Definition

  • Group of individuals: Requires at least two interacting agents ($N \ge 2$).

  • Rationality: Agents seek to maximize their payoff function. This is "common knowledge of rationality" among players.

  • Strategic behavior: Agents choose actions to maximize their payoff, anticipating rivals' moves. Payoff functions can include others' payoffs (e.g., altruism).

Main Elements in a Game

• Players: Individuals, firms, or countries interacting in the game. ($N \ge 2$).

• Strategies: A complete contingent plan of actions a player chooses in every possible situation.

  • Strategy for player $i$ is $s<em>i \in S</em>i$, where $S_i$ is the strategy set.

  • Can be discrete (e.g., $S<em>A = {6, 10}$ or $S</em>A = \mathbb{N}$) or continuous (e.g., $S<em>A = [0, 10]$ or $S</em>A = \mathbb{R}^+$).

  • Strategy sets can be symmetric ($S<em>i = S</em>j$) or asymmetric ($S<em>i \ne S</em>j$).

  • Strategy profile: A list of strategies for all players, $s = (s<em>1, s</em>2, …, s<em>N)$. Compactly denoted as $s = (s</em>i, s_{-i})$.

  • Total strategy profiles: If player $i$ has $k$ strategies and player $j$ has $m$ strategies, there are $k \times m$ profiles.

• Payoffs: The outcome (real number) a player obtains for a given strategy profile.

  • Payoff function: A mapping $u<em>i : S</em>i \times S<em>{-i} \to \mathbb{R}$, yielding $u</em>i(s<em>i, s</em>{-i})$.

Graphical Approaches for Games

• Matrices: Used for "simultaneous-move games" where players choose strategies without observing opponents.

  • Player 1 (row player), Player 2 (column player).

  • Payoffs represented as ordered pairs ($u<em>1, u</em>2$).

• Game Trees: Used for "sequential-move games" where players act one after another, often observing previous moves.

  • Start at a "root" (initial node).

  • "Terminal nodes" show payoffs at the end of a game path.

  • Strategy profile is a "path of play."

  • Information set: Connects nodes player $i$ cannot distinguish. Nodes within an information set must have the same number of actions with identical labels. A sequential game with imperfect information (via information sets) is strategically equivalent to a simultaneous-move game.

Identifying Equilibrium Behavior

• Goal: Predict stable strategy profiles ("equilibria") where no player has an incentive to unilaterally change their choice.

• Solution concepts to be studied include:

  • Deletion of strictly dominated strategies.

  • Nash equilibrium (simultaneous-move games).

  • Subgame Perfect equilibrium (sequential-move games).

  • Extensions for incomplete information (Bayesian Nash equilibrium, Perfect Bayesian equilibrium).

• ### Criteria for Evaluating Solution Concepts

  • Existence: Must find at least one equilibrium outcome.

  • Uniqueness: A single equilibrium provides precise predictions but can conflict with existence.

  • Robustness to small payoff changes: Equilibrium predictions should not change drastically with minor payoff alterations.

  • Pareto Optimality: Evaluates if other strategy profiles can make at least one player strictly better off without making any player worse off compared to the equilibrium. Equilibrium outcomes are not always Pareto optimal, often requiring coordination or rule changes to improve.