Game theory studies strategic interaction: how rational agents choose actions when outcomes depend on everyone’s choices. Core concepts:

  • Players: decision-makers.
  • Strategies: plans of action (pure or mixed/randomized).
  • Payoffs: utilities each player seeks to maximize.
  • Information: complete vs. incomplete, perfect vs. imperfect.
  • Solution concepts:
    • Nash equilibrium — each player’s strategy is a best response to others (may be multiple; can be in mixed strategies). (See Nash 1950.)
    • Dominant strategy — best regardless of others’ actions (e.g., Prisoner’s Dilemma has a dominant strategy to defect).
    • Pareto efficiency — no one can be made better off without making someone worse off.
    • Subgame perfect equilibrium — refines Nash for dynamic games (backward induction).
    • Bayesian equilibrium — for games with incomplete information (Harsanyi 1967–68).
  • Types of games:
    • Static vs. dynamic (simultaneous vs. sequential moves).
    • Zero-sum (one’s gain is another’s loss) vs. non-zero-sum (possible cooperation).
    • Cooperative vs. noncooperative (binding agreements allowed or not).
  • Classic examples:
    • Prisoner’s Dilemma — illustrates conflict between individual rationality and collective welfare.
    • Coordination games — multiple equilibria; focal points matter.
    • Hawk-Dove (Chicken) — conflict with risk of mutually bad outcome, mixed equilibria possible.
    • Ultimatum game — fairness preferences affect offers and acceptances.
  • Applications: economics (auctions, oligopoly), political science, evolutionary biology (ESS; Maynard Smith), computer science (algorithms, mechanism design), social sciences.

Key references:

  • John Nash, “Non-Cooperative Games” (1951).
  • Osborne & Rubinstein, “A Course in Game Theory” (1994).
  • Mas-Colell, Whinston & Green, “Microeconomic Theory” (1995) — chapters on game theory.

If you want, I can: (1) work through a specific example (e.g., compute Nash equilibria), (2) contrast solution concepts, or (3) show dynamic/backward-induction reasoning. Which would you prefer?

Selection in game theory often refers to how players, societies, or evolutionary processes choose among multiple possible equilibria or strategies. Selection mechanisms determine which outcome emerges when theory predicts more than one stable possibility.

Short explanation

  • Equilibrium selection: When a game has multiple Nash equilibria, selection rules or refinements (e.g., risk dominance, payoff dominance, trembling-hand perfection) help predict which equilibrium players will choose.
  • Strategy selection in evolution: Repeated interactions and mutation/replication dynamics favor strategies that are evolutionarily stable (ESS) or attractors under replicator dynamics.
  • Selection by learning: Models of adaptive learning (fictitious play, reinforcement learning) select strategies over time depending on past payoffs and information.

Examples

  1. Stag Hunt (coordination game)

    • Two equilibria: Both hunt stag (high payoff) or both hunt hare (lower but safe).
    • Payoff-dominant selection prefers stag; risk-dominant selection (players avoiding the worst-case) prefers hare.
    • Real-world: Firms choosing high-investment standard (stag) vs. safe legacy tech (hare).

  2. Hawk-Dove / Chicken (conflict game)

    • Mixed and pure equilibria exist; evolutionarily stable strategies can be mixed proportions of hawks and doves.
    • In biology: Animal territorial contests result in a stable mix of aggressive and submissive behaviors.

  3. Coordination with multiple conventions (Driving side)

    • Two pure equilibria: drive on left or right. Social conventions, laws, and historical accidents select one equilibrium and lock it in.

  4. Matching Pennies with trembling-hand refinement

    • Pure strategies aren’t stable; trembling-hand perfection rules out non-robust equilibria, selecting strategies robust to small mistakes.

  5. Rock–Paper–Scissors in evolutionary dynamics

    • No pure equilibrium; replicator dynamics can lead to cycles or a stable mixed strategy depending on payoff structure and mutation rates.

References (for further reading)

  • Harsanyi, J. C., & Selten, R. (1988). A General Theory of Equilibrium Selection in Games. MIT Press.
  • Weibull, J. W. (1995). Evolutionary Game Theory. MIT Press.
  • Young, H. P. (1998). Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton Univ. Press.

If you want, I can give a short game matrix for any of these examples or simulate selection under a chosen rule.

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