Game Theory: Nash, Sequential & Evolutionary

Learning Goal: Game Theory and Strategic Decision Making: Constructing Nash equilibria, sequential games, and evolutionary strategies in competitive systems.

Welcome to this rigorous, graduate-caliber course on Game Theory and Strategic Decision Making. This curriculum is designed to take you from a basic structural understanding of payoff mechanics to advanced mathematical modeling of dynamic, evolutionary, and informationally asymmetric competitive environments.

Prerequisites

  • Basic Probability Theory: Familiarity with expected value calculations and probability vectors.
  • Basic Calculus: Understanding of derivatives (useful for optimization in continuous strategies and evolutionary replicator dynamics).
  • Linear Algebra Basics: Matrix representation of strategic payoffs.

Course Parameters

  • Estimated Total Study Time: 15 Hours
  • Course Structure: 5 Core Modules with sequential dependencies.

Module 1: Foundations of Strategic Thinking

This module establishes the core mathematical structure of games. You will transition from conceptual descriptions of strategic conflict to formal normal-form representations. Crucially, this module directly addresses how to construct payoff matrices from real-world, text-based descriptions of games, identify dominant and dominated strategies, and mathematically formalize the iconic Prisoner’s Dilemma.

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Game theory worked example from A P Microeconomics
Game theory worked example from A P Microeconomics
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This segment introduces game theory applied to oligopolistic markets using a sandwich shop example. It covers the definition of dominant strategies, how to analyze payoff matrices, and the concept of Nash equilibrium where no firm can benefit from unilaterally changing its strategy. The analysis shows how strategic interdependence affects pricing decisions in markets with few competitors.

@khanacademy · 06:11
  • Why this video: This video directly bridges the gap highlighted in initial curriculum reviews by walking through a highly structured, step-by-step methodology for constructing payoff matrices from raw text-based business scenarios (using a localized duopoly sandwich shop example). It explicitly demonstrates how to map written strategic conditions into numerical row/column player payoffs.
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4.18 Game Theory Payoff Matrix Intro AP Micro
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4.18 Game Theory Payoff Matrix Intro AP Micro

A payoff matrix is a tool used in game theory to analyze strategic interactions between interdependent firms, where each cell displays the profits for each firm based on their strategy choices; the horizontal axis represents the first number (profit of the horizontal firm) and the vertical axis represents the second number (profit of the vertical firm), allowing analysts to evaluate different strategy combinations and determine concepts like dominant strategies, Nash equilibrium, and the prisone

@CareyLaManna · 03:35
  • Why this video: This video reinforces matrix anatomy, teaching you how to read, build, and interpret strategic interactions where firm decisions are highly interdependent. It prepares you to isolate individual player perspectives within a combined matrix cell.
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The Prisoner's Dilemma is a game theory thought experiment involving two criminals arrested and imprisoned separately. Each prisoner faces a choice: remain silent (cooperate with each other) or betray the other. The possible outcomes are: (1) Both remain silent - each serves 2 years on a lesser charge; (2) One betrays while the other remains silent - the betrayer goes free while the silent one serves 10 years; (3) Both betray each other - each serves 5 years. The dilemma arises because rational

@dannyjones · 02:44
  • Why this video: This video provides a vivid, real-world narrative on the structural incentives behind the Prisoner's Dilemma. It details why individual rational incentives persistently drive non-cooperative outcomes, illustrating the gap between joint optimization and private best responses.

Knowledge Checkpoint

  • Define the three structural elements of any normal-form game: Players, Strategies, and Payoffs.
  • Construct a 2×22 \times 2 payoff matrix from a text-based description of a competitive scenario.
  • Mathematically prove whether a player possesses a strictly dominant strategy.
  • Explain why the Nash equilibrium of a standard classic Prisoner's Dilemma is Pareto inefficient.

Module 2: Simultaneous Games & Nash Equilibrium

In this module, you will master the mathematical extraction of equilibria in simultaneous-move games. You will explore pure strategy solutions and move deep into the mechanics of mixed strategies, learning to construct best-response correspondences and solve systems of equations to find the exact probabilities that make opponents indifferent.

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5. Nash equilibrium: bad fashion and bank runs
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5. Nash equilibrium: bad fashion and bank runs

Nash Equilibrium occurs when each player's strategy is a best response to the others', meaning no one has an incentive to deviate. In coordination games like the Investment Game, multiple equilibria may exist, including Pareto-efficient and inefficient ones. Unlike Prisoner's Dilemma, communication can help coordinate on the good equilibrium without contracts or side payments, as demonstrated when a persuasive speech shifted class behavior from a bad equilibrium to a good one.

@YaleCourses · 01:09:14
  • Why this video: This classic Yale lecture offers an unmatched theoretical foundation for Nash equilibria. Using coordinating games like bank runs, it explains how self-fulfilling expectations drive equilibrium selection and explores why multiple equilibria can arise in simultaneous-choice systems.
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Mixed Strategies Nash Equilibrium: Intuition
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Mixed Strategies Nash Equilibrium: Intuition

In mixed strategies Nash equilibrium, players randomize their actions (choosing probabilities rather than fixed strategies) to make their opponents indifferent between their available choices; this prevents exploitation and creates a stable equilibrium where neither player can improve their payoff by unilaterally changing their strategy.

@AshleyHodgson · 10:18
  • Why this video: This tutorial provides the crucial intuitive jump from pure deterministic strategies to probabilistic mixing. It demystifies the core principle of mixed equilibria: a player mixes their actions precisely to make their opponent indifferent among their own pure strategies.
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Game Theory 101 (#8): The Mixed Strategy Algorithm
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Game Theory 101 (#8): The Mixed Strategy Algorithm

The process is repeated for both players. After finding player one's mixed strategy, we solve for player two's mixed strategy by calculating player one's expected utility for up and down as functions of player two's mixed strategy probability (σ_left), then setting them equal. This yields σ_left = 1/3. The complete mixed strategy Nash equilibrium requires both players to use their calculated mixed strategies simultaneously, ensuring neither can profitably deviate while the other maintains their

@Gametheory101 · 02:27
  • Why this video: A direct, algorithmic walkthrough on calculating mixed strategy Nash equilibria. It maps out the algebraic expressions of expected utility as a function of choice probabilities, ensuring you possess the core computational toolset for 2×22 \times 2 matrices.
1. Mixed Strategy Nash Equilibrium (Game Theory Playlist 4)
1. Mixed Strategy Nash Equilibrium (Game Theory Playlist 4)
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1. Mixed Strategy Nash Equilibrium (Game Theory Playlist 4)

This section concludes the analysis by finding the mixed strategy Nash equilibrium through the intersection of best response correspondences. The instructor explains that a Nash equilibrium occurs when both players' strategies are mutual best responses. In matching pennies, the only intersection point occurs at p=1/2 and q=1/2, where both players are playing mixed strategies with equal probabilities. At this point, both players are indifferent between their pure strategies and any mixture is opt

@selcukozyurt · 03:56
  • Why this video: This mathematical derivation shows you how mixed strategies emerge from the intersection of best-response correspondences. This visual and logical formulation deepens your underlying algebraic framework.

Knowledge Checkpoint

  • Determine all pure-strategy Nash equilibria in a 3×33 \times 3 matrix using the underline method.
  • Mathematically construct the algebraic equations required to make Player 1 indifferent between their available strategies.
  • Solve for the probability parameters pp and qq in a standard 2×22 \times 2 asymmetric game.
  • Explain why a player would never assign a positive probability weight to a strictly dominated strategy.

Module 3: Sequential Games & Backward Induction

Real-world strategic interactions are rarely entirely simultaneous; turn-taking and chronological positioning dictate outcomes. This module shifts focus to dynamic, extensive-form games. You will learn to map choices onto game trees, apply the solution concept of backward induction, and refine your equilibrium selection by identifying Subgame Perfect Nash Equilibria (SPNE) to eliminate non-credible threats.

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Backwards Induction Game Tree

Backwards induction is a solution method for sequential move games where you start at the end of the game tree and work backwards, eliminating dominated strategies at each decision node to determine the subgame perfect equilibrium; players make optimal choices based on anticipating future players' rational responses, which may mean they cannot achieve their most preferred outcomes even if they desire them.

@AshleyHodgson · 08:27
  • Why this video: This video teaches how to solve dynamic extensive-form game trees step-by-step. It starts from the terminal nodes and moves backward, showing how to prune branches where rational actions are dominated to determine the optimal strategic path.
5. Backward Induction, Subgame Perfect Nash Equilibrium & Nash Equilibrium (Game Theory Playlist 6)
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5. Backward Induction, Subgame Perfect Nash Equilibrium & Nash Equilibrium (Game Theory Playlist 6)

Backward induction is a solution method for extensive form games with perfect information where we start solving from the last players and work backwards, assuming sequential rationality (players choose optimally at every subgame). This process identifies the Subgame Perfect Nash Equilibrium (SPNE), which is a Nash equilibrium that remains optimal in every subgame. To verify SPNE is indeed a Nash equilibrium, we must confirm that no player can unilaterally deviate from their equilibrium strategy

@selcukozyurt · 19:14
  • Why this video: This video bridges sequential move profiles with their equivalent strategic normal form representations. It provides a formal breakdown of why certain standard Nash equilibria of a game tree fail to hold up to sequential rationality, leading to the refinement of Subgame Perfect Nash Equilibrium (SPNE).
19. Subgame perfect equilibrium: matchmaking and strategic investments
19. Subgame perfect equilibrium: matchmaking and strategic investments
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19. Subgame perfect equilibrium: matchmaking and strategic investments

Subgame perfect equilibrium refines Nash equilibrium by requiring that players' strategies constitute a Nash equilibrium in every subgame, thereby eliminating non-credible threats and ensuring consistent behavior throughout the game; this concept is essential for analyzing strategic investments because it accounts for how competitors will adjust their behavior in response to one's own decisions, which is often overlooked in simpler analyses like accounting or basic economics that ignore strategi

@YaleCourses · 01:17:09
  • Why this video: An in-depth lecture exploring subgame perfection in business settings (such as strategic investments, capacity expansions, and entry deterrence). It shows how players can manipulate game trees to make threats credible through irreversible commitments.

Knowledge Checkpoint

  • Translate a descriptive sequential-move scenario into a fully labeled game tree (extensive-form game).
  • Define the formal properties of a "subgame" inside a larger extensive-form game.
  • Apply backward induction to solve a multi-stage sequential game.
  • Identify and discard Nash equilibria that rely on non-credible threats.

Module 4: Evolutionary Game Theory & Dynamics

This module replaces the hyper-rational agents of classical game theory with biological or cultural populations governed by mutation and natural selection. You will study how strategy distributions evolve over time using the classic Hawk-Dove framework, learn the mathematical definitions of Evolutionary Stable Strategies (ESS), and explore population change using replicator dynamics.

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An Evolutionarily Stable Strategy (ESS) is a strategy that satisfies two conditions: (1) it is a Nash equilibrium, meaning no individual can gain by switching to another strategy when everyone else uses it, and (2) any rare mutant strategy that invades the population will have lower fitness than the resident strategy. The additional invasion criterion requires that for every other strategy J, the payoff of I against J must be greater than the payoff of J against itself. This ensures that even if

@ecoevotheory · 03:42
  • Why this video: This academic lecture fills a critical mathematical gap. It outlines the two strict mathematical conditions required to prove that a strategy is evolutionarily stable (ESS) when challenged by mutant strategies, laying down the core mathematical foundations of evolutionary biology models.
Evolution of aggression and sharing: the replicator dynamics with the Hawk Dove Game
Evolution of aggression and sharing: the replicator dynamics with the Hawk Dove Game
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Evolution of aggression and sharing: the replicator dynamics with the Hawk Dove Game

The Hawk-Dove game models evolutionary competition where aggressive individuals (Hawks) defeat sharers (Doves) but fight each other, while sharers share resources but lose to aggressors. Using replicator dynamics, we model population proportions X1 (sharers) and X2 (aggressors) with differential equations dX1/dt = X1(2X1 + X2 - Φ) and dX2/dt = X2(3X1 - 5), where Φ is the average fitness. When starting with mostly sharers and introducing an aggressive mutant, the population evolves toward an equi

@DrVinceKnight · 10:52
  • Why this video: This video mathematically analyzes the Hawk-Dove game using replicator dynamics. It demonstrates how to write down differential equations that represent changes in strategy frequencies within a population over time, and shows how to solve for stable polymorphic states.
12. Evolutionary stability: social convention, aggression, and cycles
12. Evolutionary stability: social convention, aggression, and cycles
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12. Evolutionary stability: social convention, aggression, and cycles

This section analyzes the hawk-dove game with general payoffs V (value of winning) and C (cost of fighting). Dove populations cannot be evolutionarily stable because hawk mutants invade successfully—hawks get V while doves get 0 when meeting doves. Hawk becomes evolutionarily stable when V ≥ C: if V > C, (H,H) is strict; if V = C, hawks still perform better against doves. This explains why species may evolve aggressive behaviors when prizes outweigh fighting costs. The analysis shows how environ

@YaleCourses · 11:24
  • Why this video: This segment uses arbitrary variables (VV for resource value, CC for fight cost) to construct a generalized payoff matrix for the Hawk-Dove game. It mathematically proves under what conditions a pure Hawk strategy fails to be an ESS, leading to a stable mixed population strategy.

Knowledge Checkpoint

  • State the two mathematical conditions required for a strategy s∗s^* to be an Evolutionary Stable Strategy (ESS).
  • Construct the generalized 2×22 \times 2 Hawk-Dove matrix using resource valuation VV and conflict cost CC.
  • Mathematically prove why a pure "Dove" population is always vulnerable to invasion by a mutant "Hawk" strategy.
  • Set up the replicator dynamics equation for a population with two competing phenotypes.

Module 5: Asymmetric Information & Auction Design

The final module explores strategic decision-making under uncertainty and uneven information. You will analyze how markets operate when one player has private information (asymmetric information), study signaling and screening models, and examine auction design. This includes analyzing bidding strategies and revenue outcomes across English and second-price (Vickrey) auctions.

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Asymmetric Information: Signaling, Unraveling & Education
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Asymmetric Information: Signaling, Unraveling & Education

In markets with asymmetric information, signals must be differentially costly across types to effectively separate them; verifiable information tends to unravel completely (informational unraveling), while unverifiable information requires costly signals that only certain types can afford, explaining phenomena like education as a signaling device and why easy-to-obtain credentials lead to qualification inflation.

@YaleCourses · 01:10:36
  • Why this video: This comprehensive Yale lecture explores asymmetric information, modeling how high-quality participants can use costly signaling (such as education) to distinguish themselves from lower-quality types and prevent market collapse.
Algorithmic Game Theory (Lecture 2: Mechanism Design Basics)
Algorithmic Game Theory (Lecture 2: Mechanism Design Basics)
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Algorithmic Game Theory (Lecture 2: Mechanism Design Basics)

In mechanism design, a second-price (Vickrey) auction achieves dominant strategy incentive compatibility, meaning bidders have no incentive to misrepresent their true valuations regardless of others' strategies; this auction format also maximizes social surplus by allocating the item to the highest-value bidder while ensuring non-negative utility for all participants, demonstrating how strategic incentives and performance guarantees can be simultaneously achieved in auction design.

@timroughgardenlectures1861 · 01:12:01
  • Why this video: This lecture covers the mathematical foundations of Mechanism Design. It provides a formal proof showing why a second-price (Vickrey) auction is dominant-strategy incentive-compatible (DSIC), proving that bidding your exact valuation is always the optimal strategy.
Auctions Overview, English and Second Price Sealed Bid
Auctions Overview, English and Second Price Sealed Bid
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Auctions Overview, English and Second Price Sealed Bid

Auctions serve as mechanisms for price discovery when sellers lack complete information about buyer valuations; the English auction is an ascending open-bid auction where the winner pays the second-highest bid plus a small increment, while the second-price sealed bid auction (Vickrey auction) requires bidders to submit private bids with the highest bidder winning but paying the second-highest bid, both formats encouraging bidders to reveal their true valuations and achieving efficient price outc

@rhill7186 · 19:48
  • Why this video: This video compares auction structures, analyzing English auctions alongside Second-Price Sealed Bid (Vickrey) mechanisms. It explores strategic bidding behavior, information update patterns, and pricing dynamics under each format.
Mechanism design theory - Eric Maskin
Mechanism design theory - Eric Maskin
@SeriousScience11:4737,110
Mechanism design theory - Eric Maskin

Mechanism design theory is the branch of economics that works backwards from desired outcomes to design institutions or rules that will achieve those outcomes, even when the designer lacks complete information about participants' preferences; a key example is the second-price sealed-bid auction mechanism, where bidders submit sealed bids and the highest bidder wins but pays the second-highest price, ensuring each participant has an incentive to bid their true valuation and thus achieving efficie

@SeriousScience · 11:47
  • Why this video: Nobel laureate Eric Maskin explains Mechanism Design theory, showing how designers can build game rules backward from a desired social outcome to align individual incentives with collective goals.

Knowledge Checkpoint

  • Explain how asymmetric information can cause adverse selection and market unraveling.
  • Define the Spence signaling condition, explaining why signals must be costly to be credible.
  • Mathematically prove why truthful bidding is a dominant strategy in a Vickrey (Second-Price) auction.
  • Compare the revenue outcomes of English auctions and Vickrey auctions under the Revenue Equivalence Theorem.

Course Map

This map outlines the structure of the course. Modules 1 through 3 build the foundational tools of strategic play, which then branch into evolutionary applications (Module 4) and mechanisms for handling asymmetric information (Module 5).

Key People Index

  • John Nash: Princeton mathematician who extended game theory beyond zero-sum games. He proved that every finite game has at least one equilibrium in pure or mixed strategies, transforming modern economic theory.
  • William Vickrey: Nobel laureate who pioneered auction theory. He designed the second-price sealed-bid auction (Vickrey auction) and developed the foundations for the Revenue Equivalence Theorem.
  • Eric Maskin: Nobel laureate in Economics who helped establish modern Mechanism Design theory, showing how to design games and systems that achieve optimal outcomes despite private information.
  • John Maynard Smith (Context from Module 4): Biologist who applied game theory to evolutionary systems. He formalized the concept of the Evolutionary Stable Strategy (ESS) and developed the Hawk-Dove game to analyze animal conflict.

Final Self-Assessment

Review this list after completing all modules. You should be able to confidently check off every item before applying these strategic models to practical scenarios.

  • I can construct a complete payoff matrix from a text-based description of a strategic situation.
  • I can find all pure strategy Nash equilibria in any n×mn \times m simultaneous game.
  • I can set up and solve the algebraic equations to calculate mixed-strategy Nash equilibria in 2×22 \times 2 games.
  • I can construct a sequential game tree from a chronological description of a game.
  • I can apply backward induction to identify the Subgame Perfect Nash Equilibrium (SPNE) in an extensive-form game.
  • I can explain why some Nash equilibria are eliminated by subgame perfection because they rely on non-credible threats.
  • I can use the two mathematical conditions of evolutionary stability to determine if a strategy is an ESS.
  • I can model population strategy shifts over time using replicator dynamics.
  • I can explain how asymmetric information leads to adverse selection and show how signaling can resolve this market failure.
  • I can mathematically prove why bidding your true valuation is the dominant strategy in a Vickrey auction.
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