Game Theory Lecture on Rationalizability and IESDS

A strategic form game lists each player’s set of possible actions and a payoff function that assigns a vector of utilities to every combination of actions. The payoff function, denoted by the lowercase symbol u, maps a strategy profile (S_1 \times \dots \times S_n) to a real‑valued vector (\mathbb{R}^n). Throughout the lecture, u represents expected utility, which incorporates beliefs about opponents and any mixed‑strategy randomization.

Rationality and Beliefs

A strategy is called reasonable when it serves as a best response to some belief about the other players’ actions. Conversely, a strategy is unreasonable if a mixed strategy strictly dominates it. Rationality therefore imposes only a minimal requirement: a player must choose a best response to some belief, without restricting what that belief can be. This weak notion of rationality leaves many strategies admissible until further knowledge is introduced.

Higher‑Order Knowledge

First‑order knowledge means each player knows that the others are rational. Second‑order knowledge adds that each player knows the others know they themselves are rational, and so on. This “crisscross” pattern of mutual rationality progressively narrows the set of plausible beliefs, sharpening the predictions that can be drawn from the model.

Formalizing Rationalizability

Rationalizability is captured by the Iterated Elimination of Strictly Dominated Strategies (IESDS). The algorithm repeatedly discards strategies that are not best responses to any belief consistent with the strategies that survived the previous round. A strategy is either a best response to some belief or strictly dominated by a mixed strategy, but never both—a result known as the theorem of equivalence. The surviving strategies after all rounds constitute the set of rationalizable strategies.

The Iterated Elimination Algorithm in Practice

1. Begin with the full strategy set (S^0).
2. In round (k+1), keep only those strategies that are best responses to beliefs that assign probability 1 to the strategies remaining after round (k).
3. Equivalently, eliminate strategies that are strictly conditionally dominated given the surviving opponent strategies.
4. Continue until no further eliminations occur.

For finite games, the algorithm must terminate because each step either removes at least one strategy or leaves the set unchanged, and the total number of strategies is limited. When the process ends with a single strategy for each player, the game is dominance solvable, delivering a unique prediction. A practical tip for problem sets is to err on the side of inclusion during the elimination steps and verify best‑response conditions only at the end.

Example: The Beauty Contest

The lecture illustrates the method with the classic beauty‑contest game. Players choose a number between 0 and 100, and each suffers a quadratic loss based on the distance between their choice and a fraction (\alpha) of the average choice. The continuous strategy space and the quadratic loss function lead the IESDS process to converge on a single rationalizable strategy: 0. This outcome demonstrates how higher‑order reasoning drives all players toward the same prediction.

Key Takeaways

• Rationalizability isolates reasonable strategies by iteratively removing those that cannot be best responses to any belief.
• Higher‑order knowledge—knowing that others are rational and that they know you are rational—tightens belief restrictions and sharpens predictions.
• The IESDS algorithm provides a formal, finite‑step procedure for identifying rationalizable strategies in any finite game.
• A game is dominance solvable when IESDS leaves exactly one strategy for each player, yielding a unique equilibrium prediction.
• In the continuous‑strategy beauty‑contest game, the algorithm converges to the unique rationalizable strategy of 0.