Introduction to Game Theory a Discovery Approach

🧭 Overview

🧠 One-sentence thesis

Game theory requires assuming that players are self-interested and perfectly logical (rational), which allows us to predict outcomes and develop strategies even though real people are not perfect.

📌 Key points (3–5)

Core assumptions: Players are self-interested (want to maximize their own payoff) and perfectly logical (always make the best decision using all available information).
What rationality includes: Each player knows the opponent is also rational and will make the best decision for themselves.
Payoff vs. winning: The goal is to maximize payoff (points, money, or anything valued), not just to "win"—a player may prefer winning by 10 points over winning by 1 point.
Common confusion: Perfect information means knowing all options and outcomes for both players, not knowing exactly how the opponent will play (the opponent still chooses within their strategy).
Why idealized assumptions matter: Like studying perfect geometric shapes, we study ideal rational players to build mathematical models, even though no one is perfect in reality.

🎯 Core assumptions about players

🎯 Self-interest

Players are self-interested: The goal is to win the most or lose the least.

Each player acts to maximize their own payoff.
Both players know the other player has the same goal.
Example (cake-cutting): Each child wants the largest piece; the cutter knows the chooser will take the largest piece, so the cutter divides evenly to guarantee a fair share.
Why it matters: Without self-interest, we cannot predict behavior—if a player doesn't like cake, they might cut themselves a small piece, leading to uneven division.

🧠 Perfect logic (rationality)

Players are perfectly logical: A player will always take into account all available information and make the decision which maximizes their payoff. This includes knowing that the opponent is also making the best decision for themselves.

A player uses all available information.
A player assumes the opponent will also choose optimally.
Example (cake-cutting): The cutter wouldn't cut one large piece hoping the chooser picks the smaller one by chance; the cutter must assume the chooser will always choose the larger piece.
Don't confuse: Rationality is an assumption for mathematical modeling, not a claim that real people are always logical.

💰 Payoff and winning

💰 What is payoff?

A player's payoff is the amount (points, money, or anything a player values) a player receives for a particular outcome of a game.

Payoff is not just "winning" or "losing"; it is a numerical measure.
The goal is to maximize payoff, which may even be negative (in which case the player wants the least negative or closest to zero payoff).

🏆 Maximizing payoff vs. simply winning

The excerpt emphasizes the difference between maximizing payoff and simply winning:

Scenario	Simply winning	Maximizing payoff
Racing	Finish first	Finish first by as large a margin as possible
Basketball	Have the higher score	Win by the largest number of points (prefer winning by 10 points over 1 point)
Election poll	Be ahead of opponent	Lead by as large a margin as possible (especially to account for polling error)

Key point: A player wouldn't just want to assure a positive payoff; they need to make sure they can't do even better with a different strategy.

🎲 Game characteristics

🔍 Perfect information

Perfect information: Each player knows what all of their own options are, what all of their opponent's options are, and both players know what the outcome of each option is. Additionally, players know that both players have all of this information.

What it includes: Knowing all options and outcomes for both players.
What it does NOT mean: Knowing exactly how the opponent will play—the opponent still makes choices.
Example: Tic-Tac-Toe and chess have perfect information; poker does not (players don't know each other's cards).
Common confusion: Perfect information ≠ knowing the opponent's exact moves in advance; it means knowing the rules, options, and consequences fully.

⏱️ Finite vs. infinite games

Finite game: The game must end after a finite number of moves or turns. In other words, the game cannot go on forever.

Infinite game: A game that is not finite. Note: An infinite game may end after a finite number of turns, but there is no maximum number of turns or process to ensure the game ends.

Finite: Tic-Tac-Toe must end after 9 or fewer turns.
Infinite: No upper limit on the number of turns, though the game might still end.

✅ Solution

A solution for a game consists of a strategy for each player and the outcome of the game when each player plays their strategy.

Example: In Tic-Tac-Toe, if both players play their best, the game will always end in a tie.

📋 Strategy definition and examples

📋 What is a strategy?

A strategy for a player is a complete way to play the game regardless of what the other player does.

The choice of what a player does may depend on the opponent, but that choice is predetermined before game play.
Example (cake-cutting):
- The cutter's strategy: Always cut evenly, no matter which piece the chooser will pick.
- The chooser's strategy: Always pick the largest piece, no matter how the cutter cuts.
Example (Tic-Tac-Toe): Player 2's strategy should determine their first move no matter what Player 1 plays first—if Player 1 plays the center square, where should Player 2 play? If Player 1 plays a corner, where should Player 2 play?

🎮 Tic-Tac-Toe as an example

The excerpt uses Tic-Tac-Toe to illustrate strategy development:

Two players: Each must decide how to play.
Perfect information: Both players know all options and outcomes.
Finite: The game ends in 9 or fewer turns.
Has a solution: If both players play their best, the game always ends in a tie.
Strategy requirement: A complete plan for every possible opponent move.

🔗 Connection to reality

🔗 Why study ideal assumptions?

The excerpt addresses the concern that no one is perfect:

Analogy to geometry: Can anyone draw a perfectly straight line or a perfectly round circle? Yet we study such objects in geometry.
Purpose: We often study ideal situations (especially in math) to build models and understand principles.
Goal: Develop strategies for perfectly logical, self-interested players as a foundation, even though real players are imperfect.