Lecture 2: Putting Yourself in Other People's Shoes

What this lecture covers

This lecture answers a question left open from the first class: what exactly are the formal ingredients of a game, and how far can dominance reasoning take you? Ben Polak introduces standard game theory notation for players, strategy sets and payoffs, then applies it to two new examples: a defender choosing which mountain pass to guard against Hannibal, and the two-thirds-of-the-average guessing game the class played previously.

After this lecture you can state the formal definition of a game, distinguish strict from weak dominance, and explain iterative deletion of dominated strategies well enough to see why the guessing game's answer keeps shrinking the more rounds of reasoning players apply. The lecture also introduces common knowledge, distinguishing it from mutual knowledge, a distinction the course will lean on again later.

Key ideas

• Formal ingredients of a game: players, each player's strategy set, and payoff functions that map a strategy profile (one strategy per player) to each player's payoff.
• Strategy profile: a specific choice of strategy for every player in the game, sometimes called a strategy vector or strategy list.
• Weak dominance: strategy s_i weakly dominates s'_i if it does at least as well against everything others might do, and strictly better against at least one thing they might do.
• Iterative deletion of dominated strategies: once you rule out a strategy as dominated, you can assume no rational opponent plays it, which can make previously undominated strategies dominated in the reduced game, and so on.
• Rationality versus knowledge of rationality: ruling out one round of dominated strategies needs only that players are rational; ruling out further rounds needs that players know others are rational, know that others know it, and so on.
• Common knowledge: a fact is commonly known if everyone knows it, everyone knows everyone knows it, and this holds for an infinite chain of "knows that" - it is stronger than mutual knowledge, where everyone simply happens to know the same fact.
• Real-world Prisoner's Dilemmas: shared projects, price competition, overfishing a common resource and carbon emissions all share the structure where individually sensible choices produce a collectively bad outcome.

Walkthrough

Recap and real-world Prisoner's Dilemmas (0:01)

Polak reviews the grade game, strict dominance and Lesson 1 (never play a dominated strategy), then extends the Prisoner's Dilemma to joint projects, price competition, overfishing and carbon emissions, arguing that communication alone cannot solve these problems: only changes to payoffs, through contracts, treaties, regulation or repeated interaction, can.

Formalizing a game: players, strategies and payoffs (6:08)

The lecture builds notation step by step: players indexed by i and j, a particular strategy s_i, the strategy set S_i, a full strategy profile s, and payoff functions U_i(s). Polak also introduces s_-i for the strategies of everyone except player i, and states the course's working assumption that strategy sets and payoffs are known to everyone, an assumption the course will later relax.

A worked abstract example (16:34)

Using a small asymmetric two-player game (Player I choosing top or bottom, Player II choosing left, center or right), the class checks for dominated strategies and confirms that center strictly dominates right for Player II, while neither of Player I's strategies dominates the other, illustrating that games need not be symmetric and dominance can apply to only one player.

Hannibal and weak dominance (24:42)

In a game where an attacker chooses an easy or hard mountain pass and a defender can guard only one, neither strategy strictly dominates for the defender, but the attacker's easy-pass strategy weakly dominates the hard pass. This introduces the formal definition of weak dominance and shows how reasoning about the opponent's weakly dominant choice lets the defender pick correctly.

The guessing game and iterative deletion (36:02)

Returning to the two-thirds-of-the-average game from the previous class, Polak walks through why choices above 67 are dominated, why choices above roughly 45 become dominated once the first group is deleted, and why this process, repeated, drives the theoretical answer toward 1. He connects each round of elimination to a deeper level of assumed rationality: your own rationality, knowledge that others are rational, knowledge that others know you are rational, and so on.

Revealing the winners and common knowledge (57:32)

The actual class average was about 13⅓, giving a target of 9, short of the theoretical limit of 1 because full common knowledge of rationality does not hold among real players. Polak closes with a hat demonstration showing that two people can each know a fact (mutual knowledge) without it being common knowledge, since neither knows that the other knows they know it.

Before you watch

• Watch Lecture 1 first for the definitions of strictly dominated strategy and the Prisoner's Dilemma, both used here without re-derivation.
• A comfort with basic algebraic notation (subscripts, functions) will help with the formal player/strategy/payoff notation, though Polak works through it slowly.

Check your understanding

1. What are the three formal ingredients needed to turn a set of outcomes into a game?
2. How does weak dominance differ from strict dominance, and how did that distinction matter for the mountain pass example?
3. Explain, step by step, why iterated deletion of dominated strategies pushes the optimal guess in the two-thirds game down toward 1.
4. Why did the actual class average land well above 1, and what does that say about common knowledge of rationality among real players?
5. What is the difference between mutual knowledge and common knowledge, and how did the hat demonstration show it?