Time Preferences in Repeated Games and Bargaining ⏳🤝

students, imagine two friends trying to split a $20 gift card. One friend says, “Let’s decide now,” while the other says, “I can wait until tomorrow.” In game theory, that difference in willingness to wait is called time preference. It matters because people do not value waiting and delaying in the same way. In bargaining, time preference can change how quickly people make offers, how long they hold out, and who ends up getting a bigger share.

Introduction: Why Waiting Changes the Game

In bargaining, a deal is often better than no deal, but not all deals are made at the same speed. If one person is impatient, delay can be costly. If both people are patient, they may wait longer to try to get a better split. This is especially important in repeated games, where players meet again and again, because waiting today can affect what happens tomorrow.

Learning objectives

By the end of this lesson, students, you should be able to:

Explain time preference in bargaining.
Describe how impatience changes outcomes.
Compare patient and impatient bargainers.

A useful idea here is discounting. A payoff received later is worth less than the same payoff received now. For example, getting $10 today may feel more valuable than getting $10 next week. In math, this is often written with a discount factor $\delta$, where $0<\delta<1$. A payoff of $x$ received one period later is valued at $\delta x$ today, and after $t$ periods it is valued at $\delta^t x$.

What Time Preference Means

Time preference is how much a person prefers a payoff now versus later. A person with high impatience strongly prefers immediate rewards. A person with low impatience, or more patience, is more willing to wait.

In bargaining, patience can change behavior in two big ways:

A patient bargainer may reject an okay offer and wait for a better one.
An impatient bargainer may accept a worse offer just to avoid delay.

Think of negotiating over who gets the last slice of pizza 🍕. If one person is hungry right now and the other is not, the hungry person may settle quickly. If both can wait, they may keep negotiating to see who gives in first.

A simple example

Suppose two players are dividing $100$. If they agree today, one gets $60$ and the other gets $40$. If they delay one day, the total still exists, but both players value it less because of waiting. If the discount factor is $\delta=0.9$, then $100$ received tomorrow is worth $90$ today.

So delay has a cost. If bargaining takes too long, the future deal becomes less attractive than a quick agreement.

How Impatience Changes Offers and Delays

Impatience affects bargaining because it changes the value of waiting. A player who is impatient loses more value from delay. That player has a stronger reason to accept an earlier offer, even if the split is not ideal.

This creates pressure in negotiations:

The more impatient side may accept smaller shares.
The more patient side can use waiting as leverage.
Delay itself becomes part of the strategy.

In many bargaining models, the side that can afford to wait longer often has an advantage. Why? Because the other side may be willing to give up more just to end the delay.

Real-world example

Imagine two companies negotiating a contract 🤝. Company A needs the deal this week to meet a deadline. Company B can wait because it has other buyers. Company A is more impatient. Since delay hurts A more, B may hold out for a better price. This does not mean B is “mean”; it means B has more patience and therefore stronger bargaining power.

Why delay can be costly for both sides

Even though one side may benefit from waiting relative to the other, delay usually wastes value overall. If a restaurant and a supplier argue for weeks about a contract, they might both lose money from missed sales. In game theory, this is why bargaining delay is important: the timing of agreement can change the final outcome, not just the division of gains.

A simple way to think about it is:

Immediate agreement gives full value.
Delayed agreement gives a discounted value.
More impatience means a bigger loss from delay.

Patient vs. Impatient Bargainers

Now let’s compare patient and impatient bargainers directly.

Patient bargainers

Patient bargainers have a higher discount factor, meaning they care more about future payoffs. They are willing to wait for a better deal because the future is still valuable to them.

Effects of patience:

They can reject weak offers.
They are less scared of delay.
They often get a larger share in bargaining situations where patience matters.

Impatient bargainers

Impatient bargainers have a lower discount factor, meaning future payoffs matter much less to them than immediate ones. They want an agreement sooner rather than later.

Effects of impatience:

They are more likely to accept smaller offers.
They dislike delay strongly.
They may concede more in order to finish negotiations quickly.

A comparison table in words

If two people are bargaining and one is patient while the other is impatient, the patient person can often wait longer. The impatient person may realize this and agree earlier. That is why patience can be a source of bargaining power.

Mini example with numbers

Suppose a player must choose between:

$50$ now, or
$60$ one period later.

If the discount factor is $\delta=0.95$, then $60$ one period later is worth $\delta\cdot 60=57$ today. In that case, waiting is better than taking $50$ now.

But if the discount factor is $\delta=0.80$, then $60$ one period later is worth $\delta\cdot 60=48$ today. Now the player would prefer $50$ now.

This shows how impatience changes decisions. A more impatient bargainer sees the future offer as less valuable and is more likely to settle immediately.

Time Preferences in Repeated Games

Repeated games are situations where players interact more than once. Time preference matters here too, because players are not just choosing one-time payoffs. They are deciding whether current actions are worth the future effects.

For example, a seller might consider cheating a customer today for a quick gain. But if the customer will return many times, the seller may lose future business. A patient seller may avoid cheating because future profits matter a lot. An impatient seller may cheat because the short-term gain feels more important.

This idea also helps explain cooperation. If players value the future highly, they are more willing to cooperate now because they expect to gain later from a good reputation or continued interaction.

Example: a trust relationship

Suppose two classmates share study notes over many weeks 📚. If one copies without sharing, that might help in the short run. But if the other stops cooperating later, both lose in the long run. A patient student understands that repeated cooperation can be better than a one-time win.

In bargaining, the same principle applies. A patient bargainer may accept a fair split today because it helps build trust and increases the chance of future agreements. An impatient bargainer may push hard for a quick advantage, even if it damages long-term relationships.

How Patience Can Shape Agreement Patterns

Time preference does not only affect who gets more. It also affects when agreement happens.

If both bargainers are patient:

They may negotiate longer.
They may be willing to wait for a better division.
They may reach agreement after several rounds.

If one bargainer is impatient:

Agreement may happen quickly.
The impatient bargainer may accept a less favorable split.
Delay becomes more threatening.

If both are very impatient:

They may rush into agreement.
They may also fail to agree if each side keeps changing demands and time runs out.

So impatience can sometimes lead to faster deals, but not always better ones. The exact outcome depends on how strong the impatience is on each side and how the bargaining process works.

A real-world negotiation example

Think about buying a used phone 📱. If the seller wants money immediately and the buyer is willing to walk away, the buyer may wait and ask for a lower price. If the seller is very patient, the seller may hold firm. If the buyer is impatient because they need the phone today, they may pay more. The final price often reflects who is more willing to wait.

Conclusion

students, time preferences are a key part of repeated games and bargaining because they show how people value the future compared with the present. A payoff today is usually worth more than the same payoff later, and this difference is captured by discounting with a factor $\delta$.

Patience gives bargainers more freedom to wait, while impatience pushes them to accept deals sooner. In bargaining, that means patient players often have more power, and impatient players often settle faster and on less favorable terms. In repeated games, patience can also encourage cooperation because future rewards matter.

Understanding time preference helps explain many everyday negotiations, from splitting chores to business contracts to deciding when to compromise. The main lesson is simple: how much you value time can change both the size of the deal and the speed of agreement ⏳🤝.

Study Notes

Time preference means how much a person values money or rewards now compared with later.
In bargaining, delay matters because future payoffs are discounted.
A common discounting idea is that a future payoff $x$ is worth $\delta x$ today, with $0<\delta<1$.
Higher patience means a higher discount factor $\delta$ and less dislike of waiting.
Lower patience means a lower discount factor $\delta$ and stronger desire for immediate payoff.
Patient bargainers can often wait for better offers.
Impatient bargainers often accept worse offers to avoid delay.
Delay can reduce the value of the final agreement for both sides.
In repeated games, patience can support cooperation because future rewards matter.
Bargaining outcomes depend not only on the size of the pie, but also on who is more willing to wait.