True story: once upon a time, Liu Bei (Emperor of Shu) and Sun Quan (Emperor of Wu) formed an ally against Cao Cao (Emperor of Wei, who had the strongest military forces at that time). However, Liu and Sun kept arguing over the ownership of Jingzhou, a state on the boundary of their countries with great strategic significance, as Liu Bei managed to “borrow” Jingzhou (in fact, only part of it), promising to return the state to Wu once he finished his expansion to the western territories.

The problem is, as Shu and Wu were not as strong as Wei, Liu and Sun had to form a coalition against Cao, otherwise neither of them was capable of fighting against Cao. On the other hand, Sun wants to take back Jingzhou before it becomes impossible to do so (as Liu’s power at Jingzhou was stabilizing). It happened that Sun was too impulsive to hold the ally and attacked Jingzhou in 219 AD. It seems that their chain of mutual distrust had inevitably ruled out the possibility of cooperation, abandoning the potential mutual benefit (of making an ally).

Similar examples abound in modern society; it is natural to wonder what the underlying reason is for people to sacrifice mutual benefit for marginal interest based on distrust. If you’re familiar with game theory, you should’ve guessed there’s a game-theoretical explanation. To unveil the secret behind this, let’s play a game:

We have learned in class about Nash Equilibrium as a solution concept of two-person nonzero-sum games. A natural extension of Nash Equilibrium in the case of dynamic games is the subgame perfect Nash Equilibrium (SPNE). A subgame of a dynamic game, stated informally, is the continuation of a game at any node in the game’s extensive form satisfying certain properties. A strategy profile (that is, a strategy combination at every step of the game) is a SPNE if it represents a Nash Equilibrium of every subgame of the original game. Intuitively, a SPNE is a strategy profile that is optimal (in the Nash sense) at any point in the game, regardless of what happened in the past.

To find a SPNE, a common method is called backward induction, which is a procedure where one first finds the optimal last move, and the other player chooses their optimal strategies accordingly. Continuing the process until the beginning of the game, one then obtains a SPNE of the game.

Remark: every SPNE must also be a Nash Equilibrium of the game, so the former is actually a stronger solution concept.

In particular, in the game, since the potential final mover is P2, we consider the last move of P2. Clearly P2 would prefer choosing “take” (with payoff 8), rather than “pass’’ (with a payoff of 7) in the last round to maximize his payoff.

Assuming P2 chooses “take” in the last round, P1 would be better off if he further chooses “take” in the previous round. Indeed, as the payoff of the game progresses linearly, it’s easy to check that continuing this process, P1 would choose “take” in the very first round. Indeed, the strategy profile where every player chooses “take” at any point in the game is the unique SPNE of the game! (Rosenthal, 1981)

As it is shown in the graph, the payoff of this game looks like a centipede. This is why this game is named as centipede game.

Thanks for playing the game, and congratulations on completing it! We make this game trying to demonstrate a peculiar Nash equilibrium result of the game interactively due to players’ mutual distrust. This resembles the renowned Prisoner's Dilemma that in both games, the players fail to reap the benefit of cooperation due to mutual distrust (in game-theoretic terms, unstable payoff pairs).

Although our game is played and studied under theoretical assumptions, it does help explain some phenomena, one of which is the Jingzhou incident back to Three Kingdoms in ancient China, which was also illustrated briefly in the introduction.
We know that Kingdom Wu had sovereignty of Jingzhou but couldn’t take it back as the alliance continues. At the same time Kingdom Shu would like to govern the city as long as possible. The emperors of both kingdoms were reluctant to keep the alliance longer than necessary, despite they both know that the longer the alliance holds, the more probable it would have been to defeat their common enemy.
One may find the situation quite similar to our centipede game. An analogy of our explanation of the game thus provides a feasible account for the controversial result that Kingdom Wu tore up the deal and betrayed Kingdom Shu to conquer Jingzhou much earlier than it could defeat Kingdom Wei on their own.

Although the result of the game seems frustrating as there is no role of trust in it, actual experiments of these games show us that few people would apply the Nash equilibrium strategies, such as those shown in the article of Rosemarie Nagel and Fang Fang Tang(1998).
All in all, when we are in a game, it would always be best to make positive communication and build mutual trust with other players to obtain a win-win solution instead of falling into invisible traps like that in the Centipede Game due to distrust.