The Infinitely Repeated Chicken Game

Introduction

In previous posts, I’ve talked about basic economic games, level-k strategic thinking, and general Nash equilibrium. One of the games I’ve covered is the Prisoner’s Dilemma (PD). I won’t re-hash it here, but if you’d like to read more about it I encourage you to read my post here. Both the prisoner’s dilemma and this article’s namesake have been used in evaluating the strategy in a Mutually Assured Destruction scenario similar to the cold war of the 20th century. The atomic bomb was cause for a very interesting theoretical discussion among economists.¹ There are not many heavily negative sum games in real life but when they do arise, the PD and Chicken Game (CG) are great for analyzing them.

Before jumping into the Chicken game, I want to add another layer of complexity to this post — repetition.

Repeated Games

In real life, most games aren’t played in a 1-shot, winner takes all. They are typically played many times and over a long period of time — repeated games.² This is the study of the different equilibria that arise in a game when it is played over and over again. Repeated games have a fixed, variable, or infinitely repeated structures. For this post, I’ll focus on an infinitely repeated game partially for simplicity and partially because I believe all games continue on forever in some capacity.

My personal favorite repeated games paper is Axelrod 1980, a computational economics paper that did simulation and AI before it was cool. The paper presents the results of a large-scale tournament in which various computerized automata were designed from scratch and pitted against one another in the repeated PD. Whichever algorithm had the best winning streak, would take home the prize. Some algorithms were vastly complex with thousands of lines and even implemented machine learning algorithms in order to slowly learn over time. Yet the winning algorithm was an exceptionally simple chunk of code called “tit-for-tat”. This strategy simply repeats the behavior of whatever the opponent did on the previous turn. If the opponent defected, defect. If they Cooperated, cooperate. Forever and always. It’s default is to cooperate if it has the first move.

Axelrod repeated the tournament in a second paper that you’ll find quite easily online. He was concerned that perhaps the nature of the participant pool made it so that the tit-for-tat algorithm was the winner. The second time around, everyone submitted all sorts of algorithms that were specifically designed to destroy tit-for-tat. The irony was that, while they did well against tit-for-tat, they got obliterated by other algorithms and thus the champion remained — tit-for-tat.

The important takeaway here is that strategies change dramatically if the game is repeated. One time, two times, or infinitely many times. Each strategy set is different depending on how many times a game is repeated. In a 1-shot PD you should defect. In a repeated game, because your opponent can counter-attack next round, you should actually cooperate (usually). Repeated games are exceptionally complicated compared to their 1-shot ancestors. But they are also exceptionally interesting.

The Chicken Game — cluck-for-cluck?

The chicken game is similar to the PD but with a slightly different payoff structure. All of these 2x2 economic games are almost entirely defined by their payout structure. And a slight deviation in the structure can change the nature and equilibrium of the game quite dramatically. The chicken game looks like this:

You can think of swerving as cooperating and going straight as defecting. The only difference here is that, unlike the prisoner’s dilemma, the payoff if you both go straight is really really bad.

Now, in a 1-shot game, this wouldn’t really matter. Who cares about -10 versus -100, I still got negative points. But what if the game is repeated? Well then I actually do care about the nominal outcome because it adds up over time. Furthermore, I have to consider what my opponent will do in retaliation. The key difference in the CG versus the PD is that the penalty for both players going straight is pretty much total annihilation. In the PD if both players confess, they get a bad outcome but not as bad as if they were a sucker.³

Equilibrium

Solving repeated games can be a bit of a headache because it gets extremely complicated to solve even a simple 2x2 game after 10, 20, or 100 repetitions. One or two repetitions can be visualized with a tree. Beyond that, one can solve these simple games analytically with the geometric discounting series (1+a+a^2+a^3…) or with a simulation to explore thousands of different strategies and their payoffs in order to find the optimal strategy for the (effectively) infinitely repeated prisoner’s dilemma. In fact, many economists, like the ones I’ve mentioned here, will use simulation or experimentation in order to solve these complex games.

In repeated games, we typically define an equilibrium based on the reasonable strategies a player could employ. These often require the player to make a threat for a future round. In other words, because the game will be repeated, players always have a credible threat for the future. Thus, if you punish me this round, I’ll punish you next round.

Logically, let’s suppose Player 1 is going to go straight on move one. Player 2 should swerve so as to not get -100. But if they do this then they will never be able to catch back up to Player 1 because player 1 already has 110 points up on them. That is, player 2 should actually decide to defect and go straight in order to level the playing field. Thus, player 1 knows that if they defect then so too will player 2 defect forever.

This is what’s known as a grim trigger strategy. If one person defects even once, it leads to defection for all eternity and everyone loses. The reverse can happen, however, if both players begin with cooperate. The threat of defecting is too large, so they cooperate for the rest of eternity. Happy days!

There is an important feature here, I haven’t mentioned and it’s a player’s ability to discount into the future. For example, if they heavily weight the outcomes in the future, their behavior might change and they might “let bygones be bygones”. But if they weight today’s outcome heavily compared to tomorrow’s, they might not be willing to let it go.

Conclusion

Repeated games are often times quite grim. One defection from either side can lead to a miserable time for both players where every single round, both players will defect, hitting back at the other until the end of time.

But if players are forward looking, and put more weight on tomorrow than today, then there can be an equilibrium in which players always cooperate.

1. Other military games include the Colonel Blotto game and it’s offshoots, first introduced in Borel 1921 and, in my view, completely defined by Kovenock and Roberson 2021. This is also an interesting, pop-take on the Colonel Blotto game. These games, while militaristic in nature, do not represent a mutually assured destruction setting like Chicken or the Prisoner’s Dilemma.

2. Probably one of my favorite papers in repeated game theory is Romero and Rosokha 2019 (yes I’m shilling Purdue professors today, deal with it). That will give you an excellent overview of what equilibrium looks like in the repeated prisoner’s dilemma using simulation. It also has some of the most gorgeous data visualizations I’ve ever seen.

3. Sucker - affectionately the outcome where you cooperate and your opponent defects.