Centipede Game

Category: Statistics
- April 16, 2025
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Analyze the classic Centipede Game from game theory. Calculate optimal strategies, payoffs at different nodes, and explore the tension between rational self-interest and cooperation in sequential decision-making.

Game Parameters

$
(0.1-0.9)

Analysis Options

About the Centipede Game

The Centipede Game is a classic sequential game in game theory that illustrates the tension between rational self-interest and mutual cooperation. Two players take turns choosing whether to "take" (end the game) or "pass" (continue playing). Each time a player passes, the total pot grows larger, but the distribution of payoffs shifts slightly toward the other player.

Game Structure

The Centipede Game has these key features:

  • Sequential Play: Players take alternating turns, starting with Player 1.
  • Growing Pot: With each pass, the total pot grows by a multiplier (typically doubles).
  • Split Ratio: At each node, the player who passes gets a smaller share of the final pot than the other player.
  • Finite Length: The game has a predetermined end point (number of moves).
The Paradox

The Centipede Game presents a fascinating paradox in game theory. Backward induction and rational self-interest suggest that the first player should take immediately at the first node. However, human players often cooperate far longer, passing multiple times to reach mutually beneficial higher payoffs.

This discrepancy between theoretical predictions and observed behavior has made the Centipede Game an important tool for studying cooperation, trust, and departures from classical equilibrium concepts in experimental economics.

Applications

The Centipede Game has applications in understanding:

  • Negotiations and bargaining behavior
  • Social dilemmas and public goods provision
  • Trust development in business relationships
  • Political standoffs and brinkmanship
  • Environmental resource management over time
Subgame Perfect Equilibrium (SPE) Formula:
Player i takes at node n if:
Ptake(i, n) > Ppass(i, n)
Where:
• Ptake(i, n) = Player i’s payoff from taking at node n
• Ppass(i, n) = Expected payoff if passing to the next move

What Is the Centipede Game?

The Centipede Game is a well-known scenario in game theory. It explores how people make decisions when they must choose between acting in their own interest or cooperating with someone else. In this game, two players take turns. Each player can either "take" a growing reward or "pass" it to the other player, letting the reward increase. However, if someone takes, the game ends immediately.

  • The pot (total reward) grows with each move.
  • The earlier a player takes, the smaller the total reward.
  • The later a player takes, the larger the reward—but only if the other player keeps passing.

Purpose of the Centipede Game Calculator

This calculator helps you explore decision-making in the Centipede Game. It shows what happens when players act logically based on payoffs and helps identify the Subgame Perfect Equilibrium, where each player chooses the best move at every step, considering what might happen later.

Whether you're a student learning game theory, a researcher in strategic decision-making, or someone curious about behavioral economics, this tool allows you to visualize the impact of rational choices and cooperative potential.

How to Use the Calculator

Follow these simple steps to get insights from the calculator:

  1. Set the Starting Amount: This is the value of the pot at the beginning of the game (e.g., $1).
  2. Choose the Growth Factor: Determines how much the pot grows each time a player passes (e.g., 2 for doubling).
  3. Set the Number of Moves: Defines how many turns the game will have (up to 20).
  4. Adjust the Split Ratio: This sets how the reward is split when a player passes (e.g., 0.8 means the passer gets 80% next round).
  5. Choose Decimal Precision and whether you want to Show Calculation Steps.
  6. Click Calculate to view results including the equilibrium point and payoff values.

What You’ll See in the Results

  • Subgame Perfect Equilibrium: Indicates at which point and by whom the game is expected to end.
  • Player Payoffs: The rewards each player receives if they follow the rational strategy.
  • Cooperative Outcome: The maximum possible payoff if both players always pass.
  • Game Table: A detailed breakdown of every node, who plays, pot value, and the optimal choice.
  • Calculation Steps: Optional detailed walk-through of how the equilibrium was determined using backward induction.

Why This Calculator Is Useful

The Centipede Game Calculator isn’t just for game theory experts. It's a practical way to understand how decisions affect outcomes in sequential scenarios. It can help with:

  • Exploring strategic decision-making
  • Studying cooperation and competition
  • Analyzing economic and psychological behavior in negotiations
  • Teaching game theory with visual and numeric clarity

This tool complements other popular analysis tools like a number sequence calculator, permutation and combination calculator, or standard deviation tool, offering a different perspective: not just finding patterns or statistics, but understanding how rational choices evolve over time.

Frequently Asked Questions

What is backward induction?

Backward induction is a method used in game theory to solve multi-step games. It starts from the end and moves backward, deciding the best move at each step assuming all future decisions will also be optimal.

Why does the rational strategy often end the game early?

Because each player considers that the next player will also act in their own self-interest, the logic moves back to the very first move. So, the first player sees no reason to pass, ending the game immediately—even if both could get more by cooperating.

What is the split ratio?

It's the percentage of the pot the player who passes receives in the next round. A higher split ratio means a better reward for passing, encouraging longer cooperation—at least theoretically.

Can this be used as a sequence solver or pattern finder?

While it’s not a typical number sequence tool, the growing pot values form a progression. You can use this as a type of sequence analysis to study how values grow over time, much like a progression tool.

Try It Out

Whether you're testing game strategies or teaching behavioral economics, this calculator gives you hands-on insight into decision logic. It's as easy to use as a mean and median tool or a probability calculator—just input your values and learn from the results.

Use it now to explore the fascinating balance of risk, reward, and rationality in sequential games.