Ultimatum Game, Backward Induction, and Stackelberg Competition

In the Ultimatum Game a proposer divides a pot while the responder decides to accept or reject the offer; a rejection gives both players zero. Ultra‑rational theory predicts acceptance of any positive amount, yet experiments show that many responders reject low offers because they perceive the split as unfair. The typical classroom version uses a $100 pot, and a simplified version uses $3 at stake. Repeated interactions reveal that current offers can shape future behavior, as players anticipate retaliation or cooperation in later rounds. The outcome highlights a limitation of Nash equilibrium: it does not capture fairness concerns or the influence of dynamic incentives in sequential settings.

Backward Induction

Backward induction solves finite extensive‑form games with perfect information by working from the final nodes toward the start. The algorithm proceeds as follows:

1. Identify the final nodes of the game tree.
2. Determine the optimal action for the player at each final node.
3. Replace those nodes with the resulting payoffs.
4. Repeat the process moving backward until the initial node is reached.

This method requires optimal play at every contingency, even at branches that never materialize in equilibrium. Consequently, it guarantees a pure‑strategy Nash equilibrium for such games and eliminates equilibria that rely on non‑credible threats—strategies that would not be optimal if actually reached.

Sequential Quantity Competition (Stackelberg)

In Stackelberg competition the first firm chooses its output before the second firm observes the choice and responds. The second firm’s best‑response function (Q_2^*(Q_1)) maps any possible quantity of the first firm to an optimal quantity for the follower. Anticipating this response, the leader selects (Q_1) to maximize profit, effectively influencing the follower’s production decision.

Compared with Cournot competition, where firms choose quantities simultaneously, the Stackelberg leader produces more. With a linear demand and constant marginal cost (C), Cournot yields each firm’s output ((1-C)/3). In Stackelberg, the leader produces ((1-C)/2) and the follower produces ((1-C)/4). The sequential move structure therefore shifts market outcomes toward the leader’s advantage.

The key insight is that “influence” matters: by moving first, a firm can shape the strategic environment and extract higher profits than in a simultaneous game.

Broader Connections

Chess exemplifies a zero‑sum game of perfect information that is theoretically solvable by backward induction, though computational limits—illustrated by references to NVIDIA’s hardware—prevent full solution in practice. The lecture underscores that information arriving over time fundamentally alters strategic analysis, a point missed when games are reduced to static strategic‑form representations.