Game Theory: Dominant Strategy & Nash Equilibrium Explained

Sequential games (extensive form games) use game trees to represent decision timelines, distinguishing them from simultaneous move games. Key components include players, available actions, information sets showing what players observe, and payoffs with the convention that the first payoff belongs to the first player. Fundamental assumptions include common knowledge of the game structure, no communication between players, rationality in maximizing payoffs, and payoffs incorporating all psychological factors. Backward induction solves these games by analyzing from the end toward the beginning: first determining optimal choices of the last-moving player, then working backward to earlier decisions. In the demonstrated example, Player B chooses between left (payoff 9) and right (payoff 1) when Player A chooses top, being indifferent. When Player A chooses bottom, Player B chooses right for payoff 1 over 0. Knowing this, Player A prefers bottom (payoff 2) over top (payoff 1). The solution is Player A chooses bottom and Player B chooses right, yielding payoffs (2,1).

Sequential games are games where players take turns making decisions, with each player observing previous actions before making their own choice. Game trees are graphical representations that show the sequence of decisions, the information available to each player at each decision point, and the resulting payoffs. Backward induction is used to solve sequential games by starting from the last decision node and working backwards, identifying the optimal action at each node given what will happen in subsequent nodes. Subgame Perfect Nash Equilibrium is a refinement of Nash Equilibrium for sequential games where a strategy profile constitutes a Nash Equilibrium in every possible subgame of the original game.

Extensive form games with perfect information are sequential games where players take turns making decisions, observing previous actions before choosing their own; the game is represented by a game tree with nodes representing decision points, histories representing the sequence of actions taken to reach each node, and terminal histories where payoffs are assigned; the player function assigns a player to each non-terminal history indicating whose turn it is to move; to find Nash equilibria, we convert the game to normal form by identifying all strategies (complete plans of action for each player at every possible decision node) and then apply backward induction to identify subgame perfect Nash equilibria, which require rational play in every subgame, eliminating non-credible threats that exist in regular Nash equilibria.

This section introduces infinitely repeated games and trigger strategies. Trigger strategies can implement cooperative outcomes: in the Prisoner's Dilemma, both players can achieve mutual denial instead of mutual confession; in Bertrand and Cournot competition, firms can share monopoly profits. The discount factor delta must exceed specific thresholds (1/4 for Prisoner's Dilemma, ~1 for Bertrand, ~1/2 for Cournot) for cooperation to be sustainable. A strategy is a complete action prescription for every possible game history. In a two-stage Prisoner's Dilemma, a strategy has 5 components. The number of strategies grows exponentially: 2^5 = 32 for two-stage Prisoner's Dilemma.

When games are played infinitely many times, cooperation becomes possible through trigger strategies. Players can induce cooperative behavior by threatening punishment: if a player deviates, the other punishes by not cooperating forever. The key question is whether the present value of always cooperating exceeds the present value of deviating once and then receiving punishment. The present value of cooperating forever is 8/(1-γ), while deviating yields 10 + 4γ/(1-γ). Cooperation is sustainable when γ ≥ 1/3, meaning players must value future payoffs sufficiently to resist short-term gains from deviation.

Grim trigger is a strategy in infinitely repeated prisoner's dilemma games where players cooperate indefinitely unless someone defects, in which case they defect forever; this strategy can sustain cooperation as a subgame perfect equilibrium when players are sufficiently patient (discount factor δ ≥ 1/2), because the threat of permanent punishment outweighs the short-term gain from defecting.

The grim trigger strategy sustains cooperation in repeated Prisoner's Dilemma by having players cooperate as long as everyone else has cooperated, but switch to permanent defection forever if anyone defects. This creates a credible punishment mechanism where any deviation triggers eternal punishment. Calculating payoffs reveals that cooperating forever yields 3/(1-β), while defecting once yields 5 + β/(1-β). Cooperation is sustainable when β ≥ 1/2—the threshold where future losses from punishment outweigh the immediate gain from defecting. This demonstrates how repeated interaction with appropriate punishment structures can sustain outcomes impossible in single-shot games.

Repeated games can sustain cooperation through future rewards and punishments, but this depends on whether the game has a known end or multiple equilibria; in finite games with a known endpoint, backward induction causes unraveling and cooperation breaks down, while in games with uncertain continuation (like infinite horizons) or those with multiple stage-game equilibria, cooperation can be sustained by using different equilibria as rewards and punishments.

A Bayesian Nash equilibrium is a solution concept for games with incomplete information where players have private types (information) and must choose strategies based on their beliefs about other players' types; to find it, one calculates expected payoffs by weighting outcomes according to the probability distribution over different game states and verifies that no player can improve their expected payoff by unilaterally changing their strategy.

Bayesian Nash Equilibrium is a solution concept for games with incomplete information where each player has private information (type) unknown to others, and players form beliefs about other players' types using probability distributions. In equilibrium, each player's strategy maximizes their expected payoff given their beliefs about others' types and others' strategies, with no player wanting to deviate. The equilibrium extends Nash Equilibrium to games with asymmetric information, where each player's strategy depends on their own type and beliefs about others' types.

Games with incomplete information are situations where players lack knowledge about certain game aspects, such as other players' payoff functions or available strategies. To handle these games, we transform them into games with perfect information by adding a 'nature' move that randomly assigns types (such as payoff functions or valuations) to players. Bayesian games are games with incomplete information where players have private information about their own types. A strategy is a complete action plan specifying what action each player will take for every possible type they might have. Expected payoffs are computed by averaging payoffs across all possible types of other players, weighted by their probabilities. A Bayesian game consists of: (1) A finite set of types for each player, (2) A probability distribution over type vectors (common knowledge), (3) Payoff functions that depend on actions and types, and (4) Conditional probabilities that allow players to update beliefs about other players' types based on their own type. Strategies are classified as: (1) Separating strategies, where different types choose different actions, allowing others to infer types from observed actions, and (2) Pooling strategies, where all types choose the same action, providing no information about types. A Bayesian Nash Equilibrium is a strategy profile where no player (and no type of any player) can improve their expected payoff by unilaterally changing their strategy. In Cournot competition with asymmetric information, firm 1 knows their own costs but not firm 2's costs (which can be high or low with known probabilities). Solving the system of best response equations yields the Bayesian Nash Equilibrium quantities: q1* = (2 - cH - cL)/3, q2H* = (1 - cH)/2, q2L* = (1 - cL)/2. In this model, firm 2 with high costs produces more than they would if their costs were known (because firm 1 must hedge against the possibility of low costs), while firm 2 with low costs produces less than they would if their costs were known (because firm 1 must hedge against the possibility of high costs).

In Bayesian games with incomplete information, players have private types with associated probabilities, and equilibrium strategies must account for beliefs about opponent types. The Bayesian Nash Equilibrium is found by determining optimal strategies for each player type: Player 1 plays Left when type is Left (payoff 1 > -2) and Right when type is Right (payoff 2 > 0), while Player 2 always chooses Y (payoff 0 > -2 and 2 > 0), regardless of Player 1's type.

Bayesian Nash Equilibrium is a game theory concept where players have incomplete information about each other's types, and it can be solved using two methods: (1) converting the game with chance nodes into a matrix form and finding Nash equilibrium, or (2) treating each type as a separate player and solving the game accordingly.

This video explains three fundamental oligopoly competition models: (1) Bertrand competition, where firms simultaneously set prices and end up producing at marginal costs (same as perfect competition); (2) Cournot competition, where firms simultaneously choose quantities, producing more than monopoly quantity but less than perfect competition; and (3) Stackelberg competition, where one firm moves first (leader) and captures more market share, producing more than Cournot but less than monopoly quantity, with the second firm (follower) producing less than Cournot equilibrium.

This section provides detailed analysis of three major oligopoly models. Cournot model: firms compete by choosing output quantities simultaneously, assuming competitor's output is fixed. Equilibrium output is n/(n+1) of perfect competition level, with price between monopoly and perfect competition. Individual firm output is 1/(n+1) of total. Bertrand model: firms compete by setting prices, assuming competitor's price is fixed. With homogeneous products and zero MC, equilibrium is P=MC (perfect competition outcome). With differentiated products, equilibrium price equals average cost. Stackelberg model: one leader moves first and knows follower's reaction function, incorporating it into profit maximization. Leader produces more than Cournot, earns higher profits. Comparison: Cournot (quantity competition, downward sloping reaction curves), Bertrand (price competition, upward sloping reaction curves), Stackelberg (leader-follower, first-mover advantage).

In a Stackelberg oligopoly with two firms producing identical goods, the first-mover (leader) maximizes profit by incorporating the second-mover's (follower) reaction function into its own profit maximization problem, resulting in the leader producing twice the output of the follower when both have identical costs; for example, with inverse demand P = 100 - Q and marginal cost of 40, the leader produces 30 units while the follower produces 15 units, yielding a market price of $55.

In oligopolistic markets with few sellers, three main models explain how firms compete: the Cournot model involves quantity-based competition where firms choose output levels based on rivals' expected production; the Bertrand model features price competition where identical products drive prices down to marginal cost, replicating perfect competition; and the Stackelberg model introduces sequential competition where a first-mover captures larger market share by committing to quantity before followers respond. These models produce different outcomes: Cournot yields prices and output between monopoly and perfect competition, Stackelberg delivers even lower prices and higher output, and Bertrand achieves competitive efficiency with zero deadweight loss. The choice of model depends on whether firms compete through quantities, prices, or timing, fundamentally shaping market efficiency and consumer welfare.

Oligopolies are classified into four types based on two criteria: whether firms choose price or quantity as their strategic variable, and whether they move simultaneously or sequentially. When firms choose price simultaneously, it's called Bertrand oligopoly; when choosing quantity simultaneously, it's Cournot oligopoly. Sequential move games are termed price leadership (when price is the choice variable) and quantity leadership (when quantity is the choice variable). These models are solved using different game theory techniques: simultaneous move games use Nash equilibrium through best response analysis, while sequential move games employ backwards induction. In quantity leadership, the second-mover's best response function is solved first, then substituted into the first-mover's maximization problem. In Cournot oligopoly, both firms' best response functions are derived and solved simultaneously to find the equilibrium quantities.