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Centipede

Game Description

The ubiquitous centipede game was first introduced in 1981 by Robert Rosenthal in his paper Games of Perfect Information, Predatory Pricing and the Chain-Store Paradox.1 This game demonstrates to participants the concept of backwards induction. It also simultaneously introduces Subgame Perfect Nash Equilibrium as a solution concept while showing participants its weakness. In the game, players are paired and given a shared resource; in this game the shared resource is a company. Each turn, a player must decide whether to cash out their investment in the company or to continue to invest. If they invest, their partner gets a more valuable company and must make the same decision. Players continue to take turns until the game reaches its end. In the last round the last player to make a decision is forced to cash out the company and gets the given division of the sum at that step.

Learning Objective 1: Backwards Induction

This game introduces participants to the concept of backwards induction. By knowing how many rounds the game will last and knowing all the possible payoffs, a participant is able to use deductive reasoning to determine what the optimal move would be in each player's turn, given what the optimal move is in the subsequent turn, starting with the last possible turn.

Learning Objective 2: Introduction of Subgame Perfect Nash Equilibrium

This game can introduce participants to the concept of Subgame Perfect Nash Equilibrium (SPNE).

Brief Instructions

Set-up


Each participant is paired with another player. They are told that as a group that they are in charge of building a company that will grow bigger (i.e. more valuable) with each player's turn.

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The Game

The first player to start has the option to either cash out the company for themselves and end the game, or to invest, letting the company grow bigger, giving their partner the same opportunity. This repeats itself until the last round in which the last mover is forced to cash out the company at the given division of the sum.

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Key Treatment Variations

Theoretical results should not vary as long as a few simple rules are followed to get the desired results.

On a practical note, the weakness of SPNE is better shown with a larger number of possible turns or steps taken. The more turns possible, the less obvious it is to players what the SPNE should be. It is recommended to configure the game with a minimum of four possible turns.

Results

This section presents the outcomes of the Centipede game across multiple groups and rounds. It showcases the distribution of player decisions, highlighting deviations from the Subgame Perfect Nash Equilibrium (SPNE) and exploring patterns of human interaction over time.

Figure 1: Aggregate Results Overview

In Figure 1, choose "Aggregate" from the dropdown menu to view the overall turn distribution across all rounds. This graph aggregates turn data and, for games with more than five groups, utilizes a box plot for clearer summarization. It allows observers to track the evolution of interactions and decision-making behaviors across rounds, illustrating how players might deviate from the theoretical predictions.

Figure 2: Game Configuration and Payoff Tree

Figure 2 displays the game configuration information, including basic repetition and participation details, alongside a detailed payoff tree and the theoretical equilibrium path predicted by SPNE. This table provides a clear reference for understanding the structured progression and optimal strategies within the game.

Figure 3: Distribution of Turns per Round

After selecting a round number from the dropdown menu, Figure 3 presents the complete distribution of turns for all groups. This graph underlines the practical challenges in achieving SPNE. A significant number of groups exceed the predicted single-step outcome, emphasizing the discrepancy between theoretical predictions and actual human behavior in strategic decision-making.

Robot Play

Our robot (i.e., an automated player) will probabilistically cash out in round one or two.

Related Content

Survivor: 21 Flags


This example from CBS's Survivor is a game that is solvable by backwards induction. After learning of backwards induction, participants can attempt to solve this game in the same way.



Solution

On your last turn, you want only one, two, or three flags remaining. Therefore, if there are four flags remaining on your turn, you will lose. If there are five, six, or seven flags remaining, you can pick up enough flags to put your partner in the situation with only four flags remaining. If there are eight flags remaining, you can only leave your opponent five, six, or seven flags, who then can put you in a position with only four flags which is a losing position. By extending this logic, you find that if you are the first player in a game with 21 flags, you have a winning strategy by working backwards.

Footnotes

1 Rosenthal, Robert W., 1981. "Games of perfect information, predatory pricing and the chain-store paradox," Journal of Economic Theory, Elsevier, vol. 25(1), pages 92-100, August.
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