Backward Induction in Game Trees: Sequential Move Game Theory

In games of imperfect information, where players may not observe previous moves, we use information sets to represent nodes a player cannot distinguish. A sub-game is a self-contained part of the game tree starting from a single node and including all its successors without breaking any information sets. Sub-game perfect equilibrium (SPE) refines Nash equilibrium by requiring that players' strategies induce a Nash equilibrium in every sub-game, eliminating implausible equilibria that rely on non-equilibrium behavior in unreached sub-games. Unlike standard Nash equilibrium, SPE ensures consistency throughout the entire game tree by applying backward induction reasoning to all sub-games.

Dynamic games are defined by players, game trees, terminal payoffs, and control assignments. Subgame Perfect Equilibrium requires optimality at every subgame, eliminating non-credible threats. Behavioral strategies assign probabilities to actions at each decision point separately, avoiding infinite-dimensional mixing. Kuhn's Theorem states that for any mixed strategy in a game with perfect recall, there exists a behavioral strategy yielding the same expected payoff. Imperfect recall occurs when players cannot remember past actions, illustrated by the 'drunk driver' example. Information sets are collections of vertices a player cannot distinguish between, used to model simultaneous games as dynamic games. Properties include: all vertices must have the same available actions, and all must be indistinguishable to the player.

In games with imperfect information, players cannot observe all previous actions. Information sets group decision nodes that a player cannot distinguish between. In extensive form representation, information sets are represented by circles around nodes. Players must choose actions that are optimal across all nodes in their information set.

An information set is a collection of nodes where a player can make a move, and the player is unaware of which specific node within that set they occupy. The player knows they are in one of the nodes but cannot distinguish which one. Trivial information sets contain only one node, meaning the player knows exactly where they are. Non-trivial information sets contain multiple nodes, representing uncertainty about the game state. This concept is fundamental for understanding games with imperfect information.

Information sets are collections of decision nodes that a player cannot distinguish between when making their choice. Within a single information set, all decision nodes must be controlled by the same player, and the action labels must be identical across all nodes in the set. Players cannot infer which specific node they are at within an information set, meaning they cannot condition their strategy on information they don't have access to. This concept allows for modeling situations where players move simultaneously or imperfectly observe previous actions, adding richness to extensive form game analysis.

Credible threats are enforceable commitments that rational opponents cannot ignore. Three conditions enable credibility: repeated play allowing future punishment, psychological manipulation creating perceived believability, or removing one's own agency to make threats unavoidable. Signaling involves communicating information to others, but signals only help if they genuinely convey useful information. In games like paper-rock-scissors, signaling doesn't work because opponents know optimal play requires randomness. Understanding when threats are credible and when signals provide value distinguishes sophisticated from naive strategic thinking.

Subgame perfect equilibrium is a solution concept for sequential games that eliminates non-credible threats by requiring all players' strategies to form Nash equilibria in every subgame; unlike standard Nash equilibrium, it ensures that threats made by players are actually believable and would be carried out if given the opportunity, thereby ruling out unsensible equilibria where players make empty threats they wouldn't follow through on.

Strategic credibility depends on the ability to implement threats when necessary. When a leader makes threats, their credibility depends on their track record of implementing similar threats. Understanding this relationship between rhetoric and action is essential for assessing the likelihood of threat implementation.

Credible describes something that can be trusted or believed. Incredible means unbelievable or astonishing. These terms are frequently tested in examinations as they relate to trust and belief.

The video explains Henry Kissinger's concept of deterrence as a psychological effect produced by a credible threat. According to this theory, strength is not in using violence but in convincing the adversary that the cost of aggression would outweigh any possible gain. For deterrence to be effective, the threat must be both credible (the adversary believes it will materialize) and effective (the action will achieve the desired outcome). The video uses the analogy of a bullying child who repeatedly threatens to smash someone's face to illustrate how ineffective threats lose their deterrent power over time. The video argues that the United States is eroding its credibility as a nation with large firepower capability by repeatedly postponing military attacks against Iran, creating doubt among adversaries about whether the US truly has the means to achieve its stated objectives.

In Stackelberg competition, firms compete over production quantities sequentially rather than simultaneously, with the leading firm choosing its quantity first and the following firm responding optimally. Using backward induction, we first derive the follower's best response function (q2 = (a - q1 - c2)/2), then substitute this into the leader's optimization problem to find the equilibrium quantities: q1 = (a + c2)/2 - c1 and q2 = a/4 - 3c2/4 + c1/2. This sequential structure results in a subgame perfect equilibrium that differs from Cournot's simultaneous-move Nash equilibrium.

The Stackelberg Model is sequential decision-making where one firm (leader) chooses output first, and the follower reacts. The leader has first-mover advantage and earns higher profits. The Price Leadership Model has one dominant firm setting price where MR = MC, with smaller firms following. Both models capture different competitive dynamics in oligopolistic markets.

The Stackelberg model extends Cournot competition by introducing sequential decision-making where one firm (leader) chooses quantity first, and the second firm (follower) observes and reacts optimally. Using the demand curve P = A - BQ with constant marginal cost C, we derive the follower's best response function q2 = (A - C)/2B - q1/2. The leader incorporates this into its own optimization, yielding q1* = (A - C)/2B and q2* = (A - C)/4B. Total quantity Q* = 3(A - C)/4B exceeds Cournot's QC = 2(A - C)/3B, resulting in lower Stackelberg prices (P* = (A + 3C)/4 < PC = (A + 2C)/3). Despite symmetric costs, the first-mover captures higher profits, demonstrating the first-mover advantage.

The Stackelberg Model is a sequential game theory model where one firm acts as a leader and another as a follower; the leader first decides its quantity, and the follower observes this decision and then optimizes its own quantity based on the leader's choice. The solution process involves: (1) deriving the follower's reaction function by maximizing its profit function with respect to its quantity, (2) substituting this reaction function into the leader's profit function, (3) maximizing the leader's profit to find the optimal quantity, and (4) substituting back to find the follower's optimal quantity. This model demonstrates how first-mover advantage can lead to higher profits for the leader compared to simultaneous decision-making models like Cournot.

The Stackelberg model is a duopoly model where firms choose quantities sequentially rather than simultaneously. Firm 1 (leader) chooses quantity first, and Firm 2 (follower) observes this choice and then decides its own quantity. Using backward induction, we first solve for the follower's best response function: Q₂ = (a - c - bQ₁)/(2b). The leader anticipates this response and substitutes it into its profit function, yielding Q₁ = (a - c)/(2b). Substituting back gives Q₂ = (a - c)/(4b). The leader produces exactly twice the follower's quantity when marginal costs are equal. The equilibrium price is P = (a + 3c)/4, and profits are π₁ = (a - c)²/(8b) and π₂ = (a - c)²/(16b), with the leader earning twice the follower's profit. This first-mover advantage allows the leader to commit to a higher quantity, forcing the follower to produce less.

This comprehensive segment introduces Perfect Bayesian Equilibrium (PBE) for games with incomplete information. PBE requires sequentially rational strategies, Bayesian belief updating, and reasonable off-path beliefs. The segment then demonstrates separating equilibrium analysis through a signaling game example: when different types choose different actions, receivers can update beliefs precisely. The instructor walks through checking whether LR' and L'R are equilibria, showing that LR' fails because Player 1 would prefer L' over R' once Player 2's strategy is determined, while L'R succeeds as both players optimize given their information sets.

Perfect Bayesian Equilibrium (PBE) extends Nash Equilibrium to games with incomplete information through four components: (1) Conditional beliefs about types, where players form probability distributions about private information; (2) Sequential rationality requiring optimal actions at every information set; (3) Consistency via Bayes' rule ensuring beliefs derive from equilibrium strategies; (4) Equilibrium conditions distinguishing on-path from off-path behavior. The instructor demonstrates PBE analysis using a gift-giving game with two player types (Friend/Enemy). The analysis covers separating equilibrium (different types choose different actions) and pooling equilibrium (different types choose same action), showing how expected utility calculations determine action thresholds (q > 1/3 for acceptance). The instructor emphasizes that equilibrium validity requires checking that no player has profitable deviation incentives.

In game theory, incomplete information refers to situations where players do not know each other's payoffs or preferences, distinguishing it from imperfect information where players don't know what others have done but do know their payoffs; unlike games with imperfect information which can be solved with existing equilibrium concepts, games with incomplete information require new solution concepts like Bayesian Nash equilibrium (for simultaneous games) and perfect Bayesian equilibrium (for sequential games) because they involve multiple types of players with different payoff structures.

In extensive form games with incomplete information, Sequential Equilibrium refines Perfect Bayesian Equilibrium by requiring that equilibrium strategies be the limit of fully mixed behavioral profiles (putting positive probability on all actions at every decision point), ensuring beliefs can be consistently computed even in unreached parts of the game tree through Bayesian updating; this approach addresses the limitation of Perfect Bayesian Equilibrium where beliefs in zero-probability information sets remain undefined.

Perfect Bayesian Equilibrium (PBE) is a solution concept for sequential games of incomplete information that requires both strategies and beliefs to be specified together, where strategies must be sequentially rational given players' beliefs and beliefs must be updated using Bayes' rule wherever possible; unlike previous equilibrium concepts that only required strategies, PBE explicitly incorporates beliefs as essential components, with sequential rationality ensuring threats remain credible based on players' updated information about the game state.