Bargaining with Absentmindedness

This paper proposes that absentmindedness, modeled as a form of bounded rationality where players cannot perfectly track game progression, serves as a novel source of bargaining power by enabling credible refusal of last-minute ultimatums, which forces preemptive concessions but ultimately leads to inefficient delays and rejected offers.

The Gist

Here is an explanation of Cole Wittbrodt's paper, "Bargaining with Absentmindedness," translated into simple, everyday language with some creative metaphors.

The Core Idea: Why Smart People Sometimes Wait Too Long

Imagine you are trying to sell your house. You and the buyer are in a negotiation. In standard economic theory, if you both know exactly how many days are left before the deal expires, you would agree immediately. Why wait? Because waiting costs money (interest, stress, etc.), and backward logic tells you that if you don't agree today, you'll just agree tomorrow for less.

But in real life, deals often drag on. People miss deadlines, get confused, or just stall.

This paper asks: What if the delay isn't because someone is being stubborn or holding out for more money, but simply because they are a little bit "absentminded"?

The author argues that forgetfulness is a superpower in a negotiation. If you are so forgetful that you don't know exactly when you are in the negotiation, you can credibly threaten to say "No" to a bad deal, even at the very last second. This forces the other person to give you a better deal earlier to avoid losing the deal entirely.

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The Metaphor: The Foggy Driver and the Sharp Navigator

To understand this, let's use the author's favorite analogy: The Absentminded Driver.

Imagine a driver (the Respondent) who is driving down a long road with many exits. They want to take a specific exit, but they are very foggy-headed. Every time they pass an exit, they forget they just passed one. They don't know if they are at Exit 1, Exit 5, or the very last Exit before the road ends.

Now, imagine a Navigator (the Proposer) sitting in the passenger seat. The Navigator knows exactly where they are. The Navigator wants the driver to take Exit A (which is good for the Navigator). The driver wants Exit B (which is good for the driver).

The Standard Scenario (Perfect Memory)

If the driver had perfect memory, the Navigator would know: "If I offer Exit A at the very last moment, the driver will say 'Yes' because they know it's the last chance."
So, the Navigator would just offer Exit A immediately. The deal happens instantly. No delay.

The Absentminded Scenario (The Fog)

But our driver is foggy. When the Navigator offers Exit A at the very last moment, the driver thinks: "Wait, am I really at the last exit? Or am I at Exit 1? If I'm at Exit 1, I should reject this bad offer and wait for a better one later."

Because the driver can't be sure, they might reject the bad offer, even if it is actually the last chance.

Here is the magic: The Navigator knows the driver is foggy. The Navigator realizes: "If I push for my preferred deal at the last minute, this foggy driver might just say 'No' and we'll both crash (get zero payoff)."

To avoid the crash, the Navigator starts offering the driver's preferred deal (Exit B) earlier in the game, just to be safe.

The Result:

1. Delay: Sometimes the Navigator tries to push for Exit A, the foggy driver says "No," and they wait another day.
2. Bargaining Power: The foggy driver gets a better deal than they would have if they were perfectly smart, simply because their confusion makes them a credible threat.
3. Inefficiency: Sometimes they wait so long that they miss the deadline entirely, and nobody gets the house.

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Key Takeaways from the Paper

1. Confusion is a Weapon

In normal games, being smart and knowing the rules is an advantage. In this specific type of negotiation, being slightly confused (absentminded) gives you power. You can credibly say, "I might reject this offer because I think we have more time," and the other person believes you.

2. The "Eleventh Hour" Threat

Usually, in a negotiation, the person making the final offer has all the power (the "Ultimatum"). But if the person receiving the offer is absentminded, they might not realize it's the "eleventh hour." They might reject a lowball offer thinking, "I'll get a better one tomorrow." The offerer, terrified of losing the deal, gives in.

3. The Cost of Being Foggy

While the absentminded player gets a better deal, the whole process is messy.

• The "Mixing" Equilibrium: The paper shows a scenario where the Navigator flips a coin. Sometimes they offer the good deal, sometimes the bad deal. The foggy driver also flips a coin: sometimes they accept, sometimes they reject.
• The Crash: Because of this coin-flipping, there is a real chance they run out of time and the deal falls through completely. The fog helps the driver get a better price, but it risks the deal dying.

4. What if Money Can Be Transferred?

The paper also looks at what happens if they can just split a pot of money (like splitting a pizza).

• If they are impatient (want money now): The "foggy" advantage disappears. The smart person can just offer a tiny bit more money to seal the deal immediately.
• If they are patient (don't mind waiting): The foggy advantage returns. They can create complex strategies where they threaten to reject offers, forcing the other person to be generous.

The Bottom Line

This paper suggests that delay in negotiations isn't always a sign of bad communication or stubbornness. Sometimes, it's a rational result of one party being unable to track the timeline perfectly.

In a world where we are all slightly distracted and forgetful, our inability to keep perfect time actually gives us leverage. We can bluff that we have more time than we do, and the other side, fearing we might walk away, will give us what we want. But the price we pay is that we sometimes miss the bus entirely.

In short: Being a little bit forgetful makes you a tougher negotiator, but it also makes you more likely to miss the deadline.

Technical Summary

Here is a detailed technical summary of the paper "Bargaining with Absentmindedness" by Cole Wittbrodt.

1. Problem Statement

Classic finite-horizon bargaining models (e.g., Rubinstein, 1982) typically predict immediate agreement due to the power of backward induction. In these models, players know the game's end and the final period's outcome, leading them to accept any offer that yields a positive payoff rather than risking a zero payoff at the deadline. However, empirical reality often shows significant delays in negotiations.

Existing literature attempts to explain delay through:

• Infinite horizons: Which often require incomplete information to generate delay.
• Incomplete information: Where players are uncertain about valuations or types.
• Reputational effects: Where players pretend to be obstinate.

The Gap: This paper addresses delay in a finite-horizon, complete information setting without relying on reputational types or private values. It introduces absentmindedness (imperfect recall) as a novel source of bargaining power and delay. The core question is: How does a player's inability to perfectly track the stage of a dynamic interaction alter the equilibrium outcome and generate inefficiency?

2. Methodology and Model Setup

The Baseline Model

• Players: A Proposer ($P$) and a Respondent ($R$).
• Actions: A binary set of actions $\{a_P, a_R\}$. $P$ prefers $a_P$, $R$ prefers $a_R$.
• Payoffs: If $a_i$ is accepted at period $t$, payoffs are $\delta^{t-1}u^i_i$. If no agreement is reached by period $T$, payoffs are 0.
• Information Structure:
  • Proposer: Has perfect recall and can condition strategies on time $t$.
  • Respondent: Is absentminded. They cannot distinguish between time periods $t=1, \dots, T$ and cannot recall the history of the game. They face all decision nodes (after receiving an offer) within a single information set.
• Strategies:
  • Proposer: A sequence of probabilities $\sigma_t$ of offering $a_P$.
  • Respondent: A single probability $p$ of accepting $a_P$ (since they cannot condition on time).

Solution Concept: Multiselves Bargaining Equilibrium (MBE)

The paper utilizes a solution concept adapted from the "Absentminded Driver" problem (Piccione & Rubinstein, 1997; Aumann et al., 1997).

• Key Feature: The respondent acts as a "single agent" making a decision at an information set that contains multiple time nodes. They maximize expected utility based on their unconditional beliefs about being at a specific node, given the strategy profile.
• Equivalence: The paper proves that an MBE is equivalent to an Ex-Ante Equilibrium, where the respondent commits ex-ante to a strategy that maximizes their expected payoff across all possible realizations of the game, treating the game as a single decision problem with random payoffs.

3. Key Results and Contributions

A. The Three Equilibria

The paper characterizes all possible equilibria for sufficiently patient players ($\delta \geq \delta(T)$):

1. Proposer-Preferred Equilibrium (No Delay):

   • $P$ always offers $a_P$; $R$ always accepts ($p=1$).
   • This mirrors the perfect recall outcome.

2. Respondent-Preferred Equilibrium (No Delay):

   • $P$ always offers $a_R$; $R$ always rejects $a_P$ ($p=0$).
   • Novelty: This equilibrium exists only under absentmindedness. Because $R$ cannot distinguish $t=1$ from $t=T$, they may rationally believe they are early in the game. If they believe the deadline is far away, rejecting $a_P$ (hoping for $a_R$ later) is optimal. $P$ concedes immediately to avoid the risk of breakdown.

3. Mixing Equilibrium (Delay):

   • Mechanism: $P$ offers $a_P$ in all periods $t < T$. In the final period $T$, $P$ mixes between $a_P$ and $a_R$.
   • Respondent's Strategy: $R$ mixes between accepting and rejecting $a_P$ with a specific probability $p^* = u^P_R / u^P_P$.
   • Logic: $R$ is indifferent between accepting $a_P$ now and waiting for the possibility of $a_R$ later. Because $R$ is absentminded, they cannot condition their rejection on the fact that it is the final period. This "rational confusion" gives $R$ credible bargaining power.
   • Outcome: There is a positive probability of delay. Deals are most likely to occur immediately before the deadline.

B. Key Properties of the Mixing Equilibrium

• Independence from Horizon: The probability of delay ($1 - p^*$) is independent of the time horizon$T$and the discount factor$\delta$(provided$\delta$ is high enough).
• Efficiency Loss: Delay is inefficient. The probability of the deal falling through entirely ($\text{Pr}(\hat{T} = \emptyset)$) increases as players become more patient ($\delta \to 1$).
  • Intuition: As players become more patient, the cost of waiting decreases. To keep the respondent indifferent between accepting and rejecting, the proposer must concede less frequently in the final period, increasing the chance of a breakdown.

C. Extensions

1. Absentminded Proposer:
If the proposer is absentminded and the respondent has perfect recall:

• Delay can still occur, but only for intermediate discount factors.
• If players are very patient ($\delta$ high), the proposer will always offer $a_P$ (even if rejected early) because they know the respondent will accept it in the final period (due to perfect recall). The "threat" of breakdown vanishes.

2. Bargaining with Side Payments (Transferable Utility):
The model is extended to bargaining over a fixed surplus $V$ (splitting the pie).

• Impatient Players ($\delta < 1$): No delay exists. The rich offer space allows the proposer to "undercut" any offer that leaves the respondent with surplus, driving the equilibrium to immediate agreement on a zero-surplus split for the respondent.
• Patient Players ($\delta = 1$): Delay equilibria exist but require non-Markovian strategies.
  • The proposer offers a "greedy" share in period 1.
  • In period 2, the proposer mixes between the greedy offer and a "fair" offer.
  • The respondent rejects the greedy offer with some probability, hoping to be in period 1 and receive the fair offer later.
  • Result: Delay exists, but unlike the baseline model, there is no efficiency loss from discounting (since $\delta=1$), though there is a risk of no deal at the deadline.

4. Significance and Implications

• Bounded Rationality as a Strategic Asset: The paper demonstrates that a cognitive limitation (absentmindedness/imperfect recall) can paradoxically increase a player's bargaining power. By being unable to track time, the respondent can credibly threaten to reject "last-minute" ultimatums, forcing the proposer to make concessions earlier.
• New Source of Delay: It provides a mechanism for delay in finite-horizon, complete-information games without relying on private information or infinite horizons.
• Efficiency Trade-off: The paper highlights a specific cost of this bargaining power: inefficiency. The very mechanism that allows the respondent to extract concessions (the credible threat of rejection due to confusion) leads to a positive probability of total deal failure.
• Relevance to Real-World Negotiations: The model offers a theoretical framework for understanding why complex, long-term negotiations (like the "Jarndyce and Jarndyce" case cited in the introduction) often stall. It suggests that when parties cannot perfectly track the complexity or timeline of a negotiation, they may adopt strategies that lead to inefficient delays and breakdowns, even when a mutually beneficial agreement is possible.

5. Conclusion

Cole Wittbrodt's paper fundamentally alters the understanding of bargaining dynamics by integrating imperfect recall. It shows that the inability to condition on the precise stage of a game breaks the logic of backward induction, creating a "rational confusion" that empowers the non-offering party. This power comes at the cost of efficiency, generating equilibria where deals are delayed or fail entirely, offering a compelling explanation for real-world bargaining friction in complex, finite-horizon environments.