Game Theory Lecture: SPNE, One-Shot Deviation, Infinite Bargaining

Backward induction remains reliable for every subgame, even when extensive‑form games contain “complicated” deviations. The logic holds because each subgame can be analyzed independently, preserving optimality from the end of the game back to the start.

Multistage Games

Multistage games consist of several stages that may be finite or infinite. Within each stage, players move simultaneously, and all actions from the previous stage are observed before the next stage begins. Typical examples include Cournot entry games, where firms decide whether to enter a market and then choose quantities, and price‑haggling scenarios that unfold over repeated rounds. The Boston game serves as a standard simultaneous‑move illustration of these dynamics.

One‑Shot Deviation Principle

A one‑shot deviation means a player changes his action at exactly one history or contingency and then returns to the original strategy thereafter.
Theorem: A strategy profile is a Subgame‑Perfect Nash Equilibrium (SPNE) if and only if no player has a profitable unilateral one‑shot deviation at any history (h).
This result simplifies verification because it eliminates the need to examine complex, multi‑history deviations. The principle requires the game to be “continuous,” a condition usually satisfied when future payoffs are discounted.

“A one‑shot deviation means I only deviate once. Or more precisely, I only deviate at one history or one contingency.”
“The upshot is that, to verify a subgame‑perfect Nash equilibrium, it’s enough to check the one‑shot deviations.”

Infinite‑Horizon Alternating‑Offer Bargaining

The bargaining model reduces negotiations to utility pairs ((x_1, x_2)) drawn from a feasible set (X) with a disagreement point at ((0,0)).
The discount factor (\delta) (where (0<\delta<1)) captures each player’s patience; a higher (\delta) indicates greater willingness to wait for future gains.

Equilibrium strategy: The proposer offers the respondent a share of (\frac{\delta}{1+\delta}); the proposer retains (\frac{1}{1+\delta}). The respondent accepts because the offer equals the discounted value of becoming the proposer in the next period.

“If the world ends tomorrow, then you as player 2 have to accept whatever I give you.”
“The first mover, player 1, is going to have an advantage in this game because they have the power of moving first.”

When (\delta \to 1) (players become very patient), the split converges to an even 50/50 division. When (\delta \to 0) (players are impatient), the proposer captures nearly the entire surplus.

Hard Facts & Numbers

• Discount factor: (\delta) ( (0<\delta<1) )
• Proposer’s share: (\frac{1}{1+\delta})
• Respondent’s share: (\frac{\delta}{1+\delta})
• Disagreement point: ((0,0))

These formulas illustrate how patience directly translates into bargaining power and how the one‑shot deviation logic underpins the verification of SPNE in both finite and infinite settings.