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 over and over. In Game Theory, we call these Repeated Games.

A game is repeated when:

Because the game repeats, you can change your moves based on what the other player did before.

Why Repeating Changes Everything

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

But when games repeat, players look ahead: "If I am nasty now, how will they treat me next time?"

This long-term view makes trust and cooperation work.

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 living
  • Compete: Drop prices low to steal the other shop's customers

If they only sold bread for one week, cutting prices makes sense. But if they sell bread every week:

  • Lowering prices today starts a price war tomorrow
  • Keeping prices fair builds a partnership that helps both for years

Strategies: Having a Plan

In repeated games, a strategy is just a rule for your next move. Here are some plans:

Tit-for-Tat: The Golden Rule of Games

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

  1. Start by cooperating. Be nice!
  2. In the next round, copy what the other player just did

This strategy works because it is:

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

The Payoff Matrix

Here are the points 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, money today is worth more than money tomorrow. We use the Greek letter δ (delta) to show how much we care about the future. It is always a number between 0 and 1.

The Math

If both players use Tit-for-Tat, they will cooperate forever. Your total score, or Total Value (V), adds up every round:

V = 3 + 3δ + 3δ2 + 3δ3 + ...

This goes on forever. In math, we call this a geometric series. When δ is less than 1, the long line of numbers simplifies beautifully to:

V = Score1 − δ

So, cooperating forever gives you a total value of:

31 − δ

What if you cheat in the first round? You get a high score of 5 right away. But Tit-for-Tat will punish you by cheating back in the next round.

If you both keep cheating, your value becomes:

V = 5 + 1δ + 1δ2 + 1δ3 + ...

The Tipping Point

Think of δ as the chance that you will play another round. Using basic algebra, we can find when Tit-for-Tat wins. It stays the best choice as long as:

δ ≥ Temptation to Cheat − Reward for CooperatingTemptation to Cheat − Punishment Score

Let's plug in our numbers:

δ
≥ 5 − 35 − 1
≥ 24
≥ 0.5

The Verdict: If there's at least a 50% chance of playing again, the math says: Cooperate!

Forgiveness

smile

Best of all, you can go back to being nice. The other Tit-for-Tat player will copy you, the punishment stops, and everyone wins again.

The Pavlov Plan

Pavlov is another strategy. Sometimes it beats Tit-for-Tat! Its rule is simple: "Win-Stay, Lose-Shift."

The math proves Pavlov fixes its own mistakes quickly.

How Pavlov Fixes Mistakes: Step-by-Step

  • Round 1: Both players cheat. They both get 1 point. That's a loss!
  • Round 2: Because they lost, both players switch. Both switch to Cooperate!
  • Round 3: They both cooperate and get 3 points. That's a win! They stay with Cooperate. Peace is restored!

Unlike Tit-for-Tat, it doesn't need a separate rule for forgiveness. The simple math of a loss forces the change.

The Big Idea

Repeated games show us that taking a smaller reward now leads to much bigger wins later. When we care about tomorrow, helping others is the smartest choice.

Where Do We See This?

Summary

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