Repeated Games: The Power of Next Time

How playing more than once changes the strategy!

In the real world people play games with each other again and again. In Game Theory, we call these Repeated Games.

A game is "repeated" when:

Because the game repeats, players can change how they act based on what others did before.

Why Repeating Changes the Game

In a one-time game, a player usually thinks: "What helps me most right now?"

But in a repeated game, players also think about the future: "If I do this now, how will others treat me next time?"

This long-term view makes cooperation and trust important.

Example: Two Shop Owners

Imagine two shop owners who sell the same bread. Every week, they choose to:

  • Cooperate: Keep prices fair so both make a steady profit
  • Compete: Lower their price to take the other person's customers

If they only sold bread for one week, lowering the price might seem best. But if they sell bread every week:

  • Lowering prices today may start a "price war" tomorrow
  • Keeping prices fair builds a partnership that helps both in the long run

Strategies: Having a Plan

In repeated games, a strategy is a rule for how to act based on what happened before. Common strategies include:

Tit-for-Tat: The Golden Rule of Games

One of the best strategies is Tit-for-Tat. It follows two simple steps:

  1. Start by cooperating (be nice!)
  2. In the next round, copy whatever the other player did last

This strategy works because it is:

To understand why Tit-for-Tat is smart, we use a Payoff Matrix and a Discount Factor.

The Payoff Matrix

Imagine the rewards for two players (You, Them):

Them
Cooperate
Defect
You
Cooperate
3, 3
0, 5
Defect
5, 0
1, 1

The Discount Factor (δ)

In math, a dollar today is worth more than a dollar tomorrow. We use the symbol δ (delta) to represent how much we value the future. It is a number between 0 and 1.

The Calculations

If both players play Tit-for-Tat, they will cooperate forever. Your total "Value" (V) is the sum of all future rounds:

V = 3 + 3δ + 3δ2 + 3δ3 + ...

However, if you decide to defect in the first round to get a higher score (5), Tit-for-Tat will punish you by defecting next round.

Grim Trigger: if this continues your value becomes:

V = 5 + 1δ + 1δ2 + 1δ3 + ...

The "Tipping Point"

Using algebra, we find that Tit-for-Tat stays the best strategy as long as:

δ ≥ Gain from defecting − Reward for CooperatingGain from defecting − Punishment Score

In our example:

δ
≥ 5 − 35 − 1
≥ 24
≥ 0.5

The Conclusion: As long as you believe there's at least a 50% chance of playing again, the math says: Cooperate!

Forgiveness

But, if you return to cooperating the other tit-for-tat will also return to cooperating and the cycle of punishment ends, and all is nice again.

Pavlov

Pavlov is another strategy, in some cases better than Tit-for-Tat. Its rule is simple: "Win=Stay, Lose=Shift."

The math shows that Pavlov is excellent at self-correcting.

If two Pavlov players accidentally defect at the same time, they both feel the "loss" (1 point) and immediately shift back to cooperating. Unlike Tit-for-Tat, it doesn't need a "forgiveness" step, the logic of the loss forces the change.

The Big Idea: Long-Term Thinking

Repeated games show us that doing a little worse now can lead to doing much better later. When we value the future, being helpful becomes the smartest way to play.

Where do we see this?

Summary

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