Game Theory Lecture: Prisoner's Dilemma to Mixed Strategy Dominance

The lecture begins by moving from informal descriptions of strategic situations to the extensive‑form representation and finally to the strategic (normal) form. This progression compresses the decision tree into a payoff matrix that captures all relevant choices and outcomes. The instructor distinguishes two modes of analysis: a positive “will play” view that predicts how rational agents actually behave, and a normative “should play” view that recommends optimal strategies.

The Prisoner's Dilemma

The classic Prisoner’s Dilemma illustrates how rational behavior can be inefficient. Two prisoners each choose either to stay mum (M) or to fink (F). The payoff matrix is:

• (M,M) = (2, 2)
• (F,M) = (3, ‑1)
• (M,F) = (‑1, 3)
• (F,F) = (0, 0)

Because F yields a higher payoff regardless of the opponent’s action, F is a dominant strategy. Both players therefore select F, producing the (0, 0) outcome, which is strictly worse than the feasible (2, 2) outcome when both cooperate. As the instructor notes, “In games, rational behavior can lead to inefficiency.”

Formalizing Strategic Behavior

The lecture introduces notation for strategy profiles (S_i) and partial profiles (S_{-i}). A strategy (S_i) strictly dominates another (S_i') if it gives a higher payoff for every opponent profile (S_{-i}). Weak dominance requires at least one strict improvement and never a lower payoff. The instructor emphasizes that multiple weakly dominant strategies cannot coexist because the strict‑inequality condition would be violated.

Beliefs and Best Responses

A belief (\beta_{-i}) is a probability distribution (lottery) over opponent strategy profiles. Expected utility is linear in these beliefs, allowing the construction of a best response: a strategy that maximizes expected utility given a specific belief. The lecture shows that several strategies can be best responses to the same belief, and graphical analysis uses the “upper envelope” of linear expected‑utility functions to identify the best‑response region.

Mixed Strategies and the Fundamental Theorem

Randomization, or mixed strategies, becomes essential when no pure strategy is a best response to any belief. The instructor cites professional poker players as experts at randomizing with precisely the right probabilities, noting that “the people who are better at randomizing with exactly the right probabilities win a lot of money.”

A strategy is called reasonable if it is a best response to some belief; otherwise it is unreasonable and is strictly dominated by a mixed strategy. The fundamental theorem presented states:

For any finite strategic‑form game, a strategy is either a best response to some belief or it is strictly dominated by a mixed strategy (but never both).

This result links dominance logic to mixed‑strategy randomization: if a pure strategy never maximizes expected utility for any belief, a suitable mixture of other strategies will dominate it.