Lecture 17: Backward Induction: Ultimatums and Bargaining

What this lecture covers

This lecture asks why backward induction, which worked cleanly for the games earlier in the course, seems to fail when real people play the ultimatum game. It starts with a live classroom experiment: Player 1 proposes a split of a pie, Player 2 can accept or reject, and rejection leaves both with nothing. Backward induction predicts Player 2 should accept almost any positive offer, yet students reject low offers and offerers tend toward a fair, even split.

From there the lecture builds a formal two-period bargaining model with discounting, then extends it to three, four, ten, and infinitely many rounds of alternating offers. By the end you can derive the classic result that patient, equally-informed bargainers converge to an even split of the pie, and you understand why real-world haggling and inefficiency arise once that model's strong assumptions are relaxed. This sits right after the course's earlier backward-induction material and sets up the following lectures on imperfect information and subgame perfection.

Key ideas

• Ultimatum game: Player 1 offers a split of a fixed pie; Player 2 accepts (both get the split) or rejects (both get zero).
• Backward induction failure in practice: experiments show people reject small positive offers and offer close to half, contradicting the "accept anything positive" prediction, likely due to pride, fairness norms, and reputation concerns.
• Dictator game: a related game where the first player simply allocates the pie with no acceptance step; people still give away a meaningful share, suggesting genuine fairness preferences.
• Discounting: money received later is worth less today, captured by a discount factor δ (e.g. δ = 0.9 means $1 tomorrow is worth $0.90 today).
• Two-stage bargaining result: the first offerer keeps 1 - δ and the other player gets δ, since the receiver can always fall back on making the next offer.
• General n-stage pattern: offers alternate in a geometric series, e.g. the four-stage offerer keeps 1 - δ + δ² - δ³, generalizing to any finite number of rounds.
• Infinite alternating-offer bargaining: as the number of rounds goes to infinity and δ approaches 1 (offers can be made rapidly), each side's share converges to 1/(1 + δ) and δ/(1 + δ), which is 50/50 when δ = 1.
• Hidden assumptions behind the even split: unlimited future offers, negligible discounting between offers, and equal patience (δ₁ = δ₂) for both players.
• Why real bargaining looks different: uncertainty about the other side's patience and about how much the pie is worth to them produces haggling, reputation-building through rejected offers, and real inefficiency, none of which appear in the full-information model.

Walkthrough

Ultimatum game and why backward induction fails (0:02)

The professor runs the ultimatum game live with several student pairs for real money, and low offers get rejected even though backward induction says any positive amount should be accepted. The class works through possible explanations: pride, comparing payoffs to the other player, wanting to teach the offerer a lesson, and the sense that a roughly equal split is fair. The dictator game is introduced as further evidence that fairness preferences, not just strategic reasoning, shape behavior even when there is no threat of rejection.

Two-stage bargaining and discounting (14:15)

The game is extended: if Player 2 rejects, roles flip and Player 2 makes an offer in a second round, but the pie shrinks by a discount factor δ to represent the cost of delay. Students play this version live and get closer to backward-induction predictions once the threat of a second round is explicit. The lecture then formalizes the result: Player 1 must offer Player 2 exactly what Player 2 could get by rejecting and moving to round two (worth δ today), keeping 1 - δ.

Extending to three, four, and ten stages (roughly 30:47)

Working backward from the last round of a three-stage game, then a four-stage game, the lecture shows the offerer's share becomes an alternating sum of powers of δ (e.g. 1 - δ + δ² - δ³ for four stages). Extending the pattern to ten stages produces a geometric series, which the class sums using the standard trick of multiplying by the common ratio and subtracting, giving a closed-form expression for the offerer's and receiver's shares.

The infinite-horizon result: the even split (around 51:18)

Letting the number of rounds go to infinity, the δ power terms vanish, leaving shares of 1/(1 + δ) and δ/(1 + δ). When offers can be made in rapid succession, δ approaches 1 and the shares converge to an even 50/50 split. The lecture stresses this striking result depends on three assumptions: potentially unlimited rounds, negligible discounting between offers, and equal discount factors (equal patience) for both players. It also predicts, unrealistically, that the very first offer is accepted with no actual haggling.

From the model to real-world bargaining (around 1:01:45)

The lecture closes by asking what the model misses: in reality, bargainers usually do not know how patient the other side is or how much the object is worth to them. This uncertainty explains why real negotiations involve back-and-forth haggling, why parties who are known to be impatient (including those with less wealth) tend to do worse, and why bargaining under incomplete information can lead to genuine economic inefficiency, including failed or delayed deals.

Before you watch

• Be comfortable with backward induction from earlier lectures on sequential games with perfect information.
• Know how to solve for equilibrium outcomes in simple take-it-or-leave-it games.
• A basic sense of discounting (money today vs. money later) is helpful, though the lecture explains it from scratch.

Check your understanding

1. Why does backward induction predict that Player 2 should accept almost any positive offer in the one-shot ultimatum game, and why does that prediction often fail in practice?
2. In the two-stage bargaining game, why must Player 1 offer Player 2 exactly δ to secure immediate acceptance?
3. How does the geometric series argument extend the two-stage result to games with many rounds, and what happens to the offerer's share as the number of rounds grows?
4. What three assumptions are needed for infinite alternating-offer bargaining to produce an even 50/50 split?
5. Why does uncertainty about the other side's patience or valuation lead to visible haggling and potential inefficiency in real bargaining, when the full-information model predicts neither?