Symmetric Games — Game Theory
An Introduction to Symmetric Games
A symmetric game is one wherein all players are on an equal footing and the same rules apply to all.
Let’s start with a simple introduction to games first. In a layman’s terms, a game is a recreational or competitive activity in which players face each other off with a set of ground rules to adhere to.
Now, game theory aims to analyze and help understand such situations where decision making is involved. Although the theory is advantageous when understanding a game better, it isn’t strictly confined to the sphere of games. Game theory comes to aid in other fields as well, such as economics, politics, and biology. A few examples of everyday situations where the theory can be nifty — firms competing for business, political candidates competing for votes, animals fighting over a hunt, and even bidders competing in an auction.
Games can be classified in many different ways. One such way is to classify games based on roles of players in the game, which results in symmetric and asymmetric games. A symmetric game is one wherein all players are on an equal footing and the same rules apply to all, whilst an asymmetric game is one where different roles and goals get assigned to every player. This article will be focusing on symmetric games.
SYMMETRIC GAMES
According to game theory, symmetric games are the kind where the payoffs for a strategy or a move depend only on the strategy employed and isn’t influenced by the player employing that strategy. This is due to all the players having the same set of strategies to choose from. The famous Prisoner’s Dilemma where both the prisoners have the same roles is a commonly discussed situation in symmetric games.
A classic example of this is Rock, Paper, and Scissors. If player A picks rock and player B picks paper, the reward is a negative one and a positive one respectively. And if the choices get reversed, the rewards interchange as well.
ZERO SUM GAMES
Games like these are also called Zero-Sum Games, wherein the sum of the rewards of all the players stays zero. A point to be noted is that not all symmetric games are zero-sum games.
Another interesting symmetrical game is Stag Hunt where two players A and B have to hunt with a stag and two rabbits available in the hunting range. Now, the stag has more meat than the rabbits combined, and for the sake of quantifying it, let’s say that the meat in the stag is 6 and the meat in a rabbit is 1. Moreover, it takes two people to hunt a stag, while a rabbit can be caught by one.
Players A and B have to bring the gear to catch either of the stag or the rabbits and have to do this without knowing what gear the other is going to bring. So if player A chooses the stag and player B chooses the rabbit, player A is going to end up with no meat while player B will win both the rabbits. But if both the players pick the stag, the meat will have to be equally divided between the two. Likewise, if both pick the rabbit, they get a rabbit each. Also, it can be seen that if the player choices were reversed, their rewards would still be the same.
Conclusion
In conclusion, symmetric games are games in which the rewards depend only on the strategies and not on the players behind the strategies. This is also the simplest way to check if a game is symmetric or not. The reader should now be able to look at a game’s payoff table and classify it as a Symmetric or Asymmetric game!
Special thanks to Mehar Jaitely for helping me out!
References:
- An Introduction to Game Theory by Martin J. Osborne
- https://en.wikipedia.org/wiki/Symmetric_game
- https://www.youtube.com/playlist?list=PLKI1h_nAkaQoDzI4xDIXzx6U2ergFmedo
That’s all for today! Stay tuned for more interesting insights on how Game Theory affects all the decisions we make.
