11.4 Sequential games and subgame perfect equilibrium

Game theory gets real when we look at sequential games. These are like chess matches where players take turns, and each move affects the next. Understanding how these games work is key to grasping strategic decision-making in the real world.

Subgame perfect equilibrium takes things up a notch. It's all about making smart choices at every step, not just overall. This concept helps us figure out why some strategies work better than others in complex, multi-step situations.

Sequential Games and Game Trees

Game Tree Structure and Elements

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• Game trees graphically represent sequential games showing order of play, decisions, and outcomes
• Nodes represent decision points while branches show possible actions or choices
• Information sets indicate when a player is unaware of previous moves
• Payoffs typically appear at terminal nodes indicating outcomes based on decision sequences
• Example: Chess game tree with initial move (e4, e5, etc.) branching to subsequent possible moves

Backward Induction Method

• Solves sequential games by reasoning backwards from end to beginning
• Process identifies optimal choices at each decision node starting from final moves
• Used to find subgame perfect equilibria by determining optimal strategy at every decision point
• Example: Solving ultimatum game by first considering responder's optimal choice, then proposer's

Subgame Perfect Equilibrium

Concept and Refinement of Nash Equilibrium

• Subgame perfect equilibrium (SPE) refines Nash equilibrium for sequential games
• Requires optimal play in every subgame of the original game
• Subgame defined as any part of game tree considered as separate game from single decision node
• SPE eliminates non-credible threats by ensuring strategies optimal at every decision point
• Addresses Nash equilibrium limitation in sequential games where some rely on non-credible strategies
• Example: Centipede game where SPE differs from some Nash equilibria relying on non-credible threats

Finding and Analyzing SPE

• Identify all subgames within original game
• Solve for Nash equilibria in each subgame working backwards to game start
• Number of SPE always less than or equal to Nash equilibria count
• SPE unique in perfect information games solved by backward induction (no payoff ties)
• Example: Solving entry deterrence game by first analyzing subgame after potential entry, then working backwards

Credibility in Sequential Games

Assessing Threat and Promise Credibility

• Credible threats/promises carried out if conditions occur non-credible ones not in player's best interest
• Time consistency crucial requiring strategy remain optimal as game progresses
• SPE identifies credible threats/promises by ensuring optimal strategies at all decision points
• Chain-store paradox illustrates how rational one-shot game threats become non-credible in sequential context
• Example: Analyzing credibility of incumbent firm's threat to engage in predatory pricing against new entrants

Enhancing Credibility in Games

• Commitment devices make non-credible threats/promises credible by limiting future options or altering incentives
• Reputation effects in repeated games enhance credibility by creating long-term follow-through incentives
• Game-theoretic analysis compares short-term deviation benefits with long-term lost credibility consequences
• Example: Firm building excess capacity as commitment device to deter market entry by competitors