Game Theory Overview 01/24/2026 14:06:16 Game theory explained Game Theory is a branch of mathematics and economics that studies... give me game theory problems Alright 😈 Here are game theory problems, starting from classic → hard ... What ?, What ?, Id don't understand, Can y you give me FULL course of... Game Theory — Full Beginner-to-advanced Course Copy Edit Download... I don't want it crash course,, give me lesson 1 Perfect 👍 We’ll go slow, clear, and only Lesson 1. No rush, no big words.... lesson 2 Good 👍 We continue slowly, clearly, and only Lesson 2. Lesson 2: Payoff... Lesson 3 Nice, you’re doing this the right way 👍 Slow, one lesson at a time. Less... I don't understand Pure Strategies and Mixed Strategies well That’s totally okay 👍 This is the most confusing part for beginners, so... Give me the solution of the exercise you gave me the past message Got it 👍 I’ll solve the exercise from my last message (the Tiny mental... I meant this one Perfect 👍 thanks for clarifying. Let’s solve that exact exercise, slowly an... Lesson 4 Great 👍 Lesson 4 is one of the MOST important lessons in game theory.... Check my answers in the last test: 1- Yes, L is dominant Good — let’s check it carefully, step by step, to be 100% sure. 🔹 The... Lesson 5 Excellent 🔥 Lesson 5 is the CORE of game theory. Take it slow — once t... Check my answers: 1. A:R 2. B:R 3. Yes, (R, R) Great — let’s check each answer carefully, one by one. You’re very close... Lesson 6 🚀 Powered by ChatGPT Exporter 1 / 147 Excellent 🔥 Lesson 6 builds directly on what you already know. We’ll go... Check my answers: 1- Yes 2- A:R eliminated 3- Yes 4- (L, L) Excellent — let’s check each step carefully. Short answer first: you are... Lesson 7 Perfect. Lesson 7 is where mixed strategies finally make SENSE. We go... Check my answers: 1- No, 2- To make the enemies nonsense from them Nice attempt 👍 Your idea is right, we just need to polish the wording to... Lesson 8 Let’s go 🔥 Lesson 8 = the moment mixed strategies become mechanica... Check my answers: 1. No 2. Nash Equilibria: - A:(U=2/3,D=1/3) - B:... Alright, let’s check them carefully — step by step, no mercy 😄 🧠 🔹 The... Yes I make it 1 + P or Q instead of 1 - P or Q, Sorry for Wrong YES — that explains it perfectly 😄 👏 And honestly? That’s a classic... Lesson 9 Alright 😄 Lesson 9 = what players actually get from the game. Not... Check my answers: 1. No 2. Equilibria: - A(U=1/2, D=1/2) - B(L=1/2,... Great — let’s check them properly, one by one 👀 🧠 Short version: you g... Sorry for negative numbers and operation multiplication error, Lesson 10 No worries at all — that kind of slip happens to everyone, especially onc... Lesson 11 Alright, buckle up 😄 Lesson 11 is a BIG one — this is where repeated... Lesson 12 Alright 😄 Lesson 12 = we leave tables and enter game trees. This is... Lesson 13 Alright 😄 Lesson 13 is where uncertainty enters the game. Players don’t... Lesson 14 Let’s go 😄 Lesson 14 = we make signaling precise and deadly accurate.... Lesson 15 Alright 😄 Lesson 15 flips signaling upside-down. Now the uninformed... Lesson 16 Alright 😄 Lesson 16 is the crown jewel. Here we stop analyzing games a... 🚀 Powered by ChatGPT Exporter 2 / 147 Give me full exam LET’S GO 😈 🧠 Here is a FULL GAME THEORY EXAM, exactly like a... Summary Sheet for revision Perfect choice 👍 Here is a ONE-PAGE-STYLE SUMMARY SHEET — clean,... Ultra-compact formula sheet Alright, here’s an ULTRA-COMPACT Game Theory formula sheet — zero... Give me full exam of the mistakes i made Got you 👍 Here is a FULL PRACTICE EXAM custom-built from your exact... 🔥 turn it into a timed mock test (With REAL Timer, Choose Questions... 🔥🔥 LET’S DO IT — REAL TIMED MOCK EXAM (Bubble / MCQ style) 🔥🔥 ... 1- B 2- B 3- C 4- A 5- C 6- B 7- A 8- B 9- C 10- B 11- C 12- B 13- D 14- C 1... 🔥 EXAM RESULTS — TIMED MOCK TEST 🔥 Nice discipline submitting... END OF CHAT, Say "Goodbye!" Goodbye! 👋 😄 🚀 Powered by ChatGPT Exporter 3 / 147 Game theory explained Game Theory is a branch of mathematics and economics that studies strategic decision-making — situations where the outcome of your choice depends not only on what you do, but also on what others do. Think of it as: “I choose my move while thinking about what you will choose.” 1. Core idea (very simple) A game in game theory has: 1. Players – decision makers 2. Strategies – choices available to each player 3. Payoffs – results (win/lose, profit/loss, reward) 4. Rules – who moves when and what is allowed Each player tries to maximize their payoff 2. Classic example: Prisoner’s Dilemma Two suspects are arrested and questioned separately. Prisoner B: Silent Prisoner B: Betray Prisoner A: Silent A: 1 year, B: 1 year A: 10 years, B: 0 Prisoner A: Betray A: 0, B: 10 years A: 5 years, B: 5 years Key insight: Betraying is the rational choice for each prisoner individually. But both betraying leads to a worse outcome than mutual silence. This shows how individual rationality can lead to a bad collective result 🚀 Powered by ChatGPT Exporter 4 / 147 3. Nash Equilibrium (VERY important) A Nash Equilibrium is a situation where: No player can improve their payoff by changing their strategy alone In Prisoner’s Dilemma: Both betray = Nash equilibrium Even though it’s not the best overall outcome. 📌 Many games have multiple equilibria or none at all. 4. Types of games Cooperative vs Non-Cooperative Cooperative : Players can make binding agreements Non-Cooperative : No trust, no enforcement (most famous models) Zero-Sum vs Non-Zero-Sum Zero-sum : One player’s gain = another’s loss Example: Chess, Poker (simplified) Non-zero-sum : Everyone can win or lose together Example: Trade, climate agreements Simultaneous vs Sequential Simultaneous : Players move at the same time Sequential : Players move one after another (think chess) 5. Dominant strategies A dominant strategy is one that is best no matter what others do Example: In Prisoner’s Dilemma, betray is dominant. If a player has a dominant strategy → they will use it. 🚀 Powered by ChatGPT Exporter 5 / 147 6. Mixed strategies Sometimes the best strategy is to randomize Example: Rock–Paper–Scissors Best strategy = play each move with probability 1/3. This prevents opponents from exploiting patterns. 7. Real-world applications 🎮 Games & AI NPC behavior Competitive matchmaking AlphaGo-style decision making 💰 Economics Pricing wars between companies Auctions (eBay, ad bidding) Market competition ⚔ Politics & War Nuclear deterrence Cold War strategies Voting systems 🧠 Biology Evolutionary strategies Animal behavior Survival competition 8. Evolutionary Game Theory (bonus) Strategies evolve over time based on success. 🚀 Powered by ChatGPT Exporter 6 / 147 “Survival of the best strategy” Used in biology and AI training Example: Cooperation can evolve if repeated interactions exist. 9. Why game theory is powerful It explains: Why smart people make “bad” outcomes Why cooperation is hard Why threats, promises, and trust matter It is logic under conflict If you want, I can: Explain it with code Connect it to AI or game development (Godot, Roblox, etc.) Give hard problems or real war/game simulations Explain minimax , backward induction , or mechanism design Just tell me 🔥 🚀 Powered by ChatGPT Exporter 7 / 147 give me game theory problems Alright 😈 Here are game theory problems , starting from classic → hard → evil-tier NO solutions included (unless you ask). 🟢 Level 1 – Classic & Fundamental Problem 1: Prisoner’s Dilemma (Warm-up) Two players simultaneously choose Cooperate (C) or Defect (D) C D C (3,3) (0,5) D (5,0) (1,1) Tasks: 1. Find all Nash equilibria 2. Identify dominant strategies 3. Is the equilibrium Pareto optimal ? Problem 2: Battle of the Sexes Two players want to meet but prefer different events. Opera Football Opera (2,1) (0,0) Football (0,0) (1,2) Tasks: 1. Find all pure strategy Nash equilibria 2. Does a mixed equilibrium exist? 🚀 Powered by ChatGPT Exporter 8 / 147 3. Is coordination guaranteed? 🟡 Level 2 – Strategic Reasoning Problem 3: Dominance Elimination Player A chooses rows, Player B chooses columns. L M R T (2,1) (1,0) (0,2) M (3,2) (2,1) (1,0) B (1,3) (0,2) (2,1) Tasks: 1. Eliminate strictly dominated strategies 2. Does the process lead to a unique outcome ? 3. Is the result a Nash equilibrium? Problem 4: Mixed Strategy Game Zero-sum game: L R U 2 −1 D −2 1 Tasks: 1. Does a pure Nash equilibrium exist? 2. Find the mixed strategy equilibrium 3. Compute the value of the game 🔴 Level 3 – Sequential Games (Hard) Problem 5: Entry Deterrence An Incumbent firm (I) moves first. 🚀 Powered by ChatGPT Exporter 9 / 147 Chooses Fight or Accommodate Then Entrant (E) chooses Enter or Stay Out Payoffs: Stay Out → (I: 3, E: 0) Enter + Fight → (−1, −1) Enter + Accommodate → (1, 2) Tasks: 1. Draw the game tree 2. Solve using backward induction 3. Find the Subgame Perfect Nash Equilibrium Problem 6: Ultimatum Game Player 1 offers how to split $100 Player 2 accepts or rejects Accept → split as proposed Reject → both get 0 Tasks: 1. Predict outcome using rational agents 2. Compare with real human behavior 3. Is the equilibrium socially efficient? ⚫ Level 4 – Expert / Evil Tier 😈 Problem 7: Repeated Prisoner’s Dilemma The Prisoner’s Dilemma is played infinitely with discount factor Tasks: 1. For which values of can cooperation be sustained ? 2. Analyze Grim Trigger vs Tit-for-Tat 3. Does the Folk Theorem apply? δ ∈ (0, 1) δ 🚀 Powered by ChatGPT Exporter 10 / 147 Problem 8: Bayesian Game (Incomplete Information) Player A may be Strong or Weak with probability Player B observes the action but not the type. Tasks: 1. Define strategies and beliefs 2. Find Bayesian Nash equilibria 3. Determine if signaling occurs Problem 9: Mechanism Design A seller wants to auction an item. Buyers have private valuations Tasks: 1. Design a mechanism that is: Incentive compatible Individually rational 2. Compare First-Price vs Second-Price auctions 3. Maximize seller revenue ☠ Final Boss Problem (Research-Level) Problem 10: Evolutionary Stability Given population strategies with payoff matrix: S₁ S₂ S₁ 3 1 S₂ 4 2 Tasks: 1. Find all Nash equilibria 2. Identify ESS (Evolutionarily Stable Strategies) 3. Analyze replicator dynamics p S , S 1 2 🚀 Powered by ChatGPT Exporter 11 / 147 If you want: ✅ Full solutions 💻 Python simulations 🎮 Game-dev applications (AI opponents, minimax bots) ⚔ War / Cold War strategy modeling Say the word and I’ll go full professor mode 😤 📚 🚀 Powered by ChatGPT Exporter 12 / 147 What ?, What ?, Id don't understand, Can y you give me FULL course of game theory ? Game Theory — Full Beginner- to-Advanced Course This course assumes no prior knowledge . We start from zero and build step by step. Lesson 0: What Is Game Theory? Game Theory studies situations where your outcome depends on your action AND others’ actions Examples: Two companies setting prices Two countries threatening war Two players choosing moves in a game Key idea: Think before acting, because others are thinking too. Game Theory — Full Beginner-to-advanced Course Copy Edit Download 🚀 Powered by ChatGPT Exporter 13 / 147 Lesson 1: What Is a Game? Every game has four parts: 1. Players – decision makers 2. Strategies – choices players can make 3. Payoffs – rewards or losses 4. Rules – order of moves and information Example Two players choose Left (L) or Right (R) If both choose the same → both win If different → both lose This is already a game. Lesson 2: Payoff Matrices We represent games using tables. Example: B: L B: R A: L (2,2) (0,0) A: R (0,0) (1,1) (A’s payoff, B’s payoff) 🚀 Powered by ChatGPT Exporter 14 / 147 Lesson 3: Strategies Pure Strategy Choosing one fixed action (always L or always R) Mixed Strategy Randomizing between actions Example: L with 50%, R with 50% Used when being predictable is bad. Lesson 4: Dominant Strategies A strategy is dominant if it is always better no matter what others do If a player has a dominant strategy: They will always choose it Rational players never ignore dominant strategies Lesson 5: Prisoner’s Dilemma (VERY IMPORTANT) Cooperate Defect Cooperate (3,3) (0,5) Defect (5,0) (1,1) Observations: 🚀 Powered by ChatGPT Exporter 15 / 147 Defect is dominant Both defect → bad outcome Lesson: Rational choices can lead to bad collective results. Lesson 6: Nash Equilibrium A Nash Equilibrium is a situation where: No player can improve their payoff by changing strategy alone Key facts: Every finite game has at least one Nash equilibrium Can be pure or mixed Prisoner’s Dilemma equilibrium: (Defect, Defect) Lesson 7: Eliminating Dominated Strategies Steps: 1. Remove strategies that are always worse 2. Reduce the game 3. Repeat This often reveals the equilibrium. 🚀 Powered by ChatGPT Exporter 16 / 147 Lesson 8: Zero-Sum Games In zero-sum games: One player’s gain = other’s loss Examples: Chess Rock–Paper–Scissors Key concept: Minimax Minimize maximum possible loss Lesson 9: Mixed Strategy Equilibrium Used when no pure equilibrium exists. Example: Rock–Paper–Scissors Best strategy: 1/3 each Why? Prevents exploitation Lesson 10: Sequential Games Players move one after another Represented using game trees 🚀 Powered by ChatGPT Exporter 17 / 147 Key method: Backward Induction Solve from the end Move backwards Lesson 11: Subgame Perfect Nash Equilibrium A strategy must be optimal: In the full game In every subgame Eliminates non-credible threats. Lesson 12: Repeated Games Games played many times. Important result: Cooperation can emerge if players care about the future Concepts: Discount factor Punishment strategies Lesson 13: Bayesian Games (Incomplete Information) 🚀 Powered by ChatGPT Exporter 18 / 147 Players lack full information. Examples: Auctions Poker Military deception Key ideas: Types Beliefs Bayesian Nash Equilibrium Lesson 14: Auctions & Mechanism Design Auction types: First-price Second-price (Vickrey) Mechanism design goal: Design rules so truth-telling is optimal Used in: Google Ads Online markets Lesson 15: Evolutionary Game Theory 🚀 Powered by ChatGPT Exporter 19 / 147 Strategies evolve over time. Key ideas: Fitness Replicator dynamics ESS (Evolutionarily Stable Strategy) Used in: Biology AI training Lesson 16: Game Theory in AI & Programming Applications: NPC decision-making Competitive bots Reinforcement learning Used in: Chess engines RTS games Economic simulations What You Should Do Next 1. Solve small payoff matrices 🚀 Powered by ChatGPT Exporter 20 / 147