Karma — A Mathematical Proof

Photo Credit: Kyle Ellefson (Unsplash)

Karma. The Golden Rule. Whatever you want to call it. There’s the general idea shared across many religions and philosophies that we should do unto others as we want to be done unto us, because what goes around comes around.

In real life, the rule tends to hold true, people who are good and treat others well, generally have good outcomes. But, how is this rule enforced and why doesn’t always hold true there’s no universal class captain maintaining a ledger of people’s deeds and making sure that bad people get bad outcomes.

However, mathematics, yes maths, has a theory about karma that explains why it tends to work and when relying on karma is good strategy.

Let me introduce you to the prisoners dilemma. Imagine when Dammy Krane and his manager where arrested on a flight that was booked with stolen credit card details.

You’re Dammy Krane, the police take you to a small room, your manager is being interrogated in a different room. An officer lays out your options:

1. If you confess to stealing the card and your manager doesn’t confess. You get a small fine while he gets 5 years.
2. If you confess and turn on your manager but he also does the same. Both of you get 3 years.
3. If both of you choose not to confess, you both get 1 year in prison.
4. While if your manager turns on you, but you don’t confess. He gets a small fine, while you get 5 years

In this situation, the optimal strategy for Dammy Krane is to rat on his manager. If his manager keeps quiet, he gets just a small fine, if his manager rats on him, he gets 3 years as opposed to the 5 he would have gotten if he kept quiet.

Now, imagine this game is played multiple times with different players using different strategies, what do you think the best strategy would be? In this situation the strategy is to keep quiet the first time and then on your next turn, you should match whatever action your opponent took on his last turn.

This strategy was proven to be optimal by Robert Axelrod, a political science professor at University of Michigan, he set up a repeated game with the rules of the prisoners dilemma and invited mathematicians and computer scientists to submit strategies to win this game. After the first round, when the “karma” strategy won, he still opened up submission to people from all around the world and even with the knowledge of the winning strategy from the first round, no one was able to design a better one.

Why is a different strategy optimal in a repeated game as opposed to a single game. Well, because of karma, in a repeated game, other people can see your previous moves and will react accordingly.

Most of real life is a repeated game, people can see your previous moves and would react accordingly, so those that follow the golden rule would tend to have optimal outcomes.

But not all of life is a repeated game and in some instances, it is best to optimise for only yourself when making decisions. In his paper, Axelrod gives for rules for when the Golden Rule is optimal strategy

1. It must be a repeated game, in one off encounters, it may be reasonable to optimise for only yourself
2. You can’t be the only nice person, there has to be a chunk of people willing to reciprocate good behaviour. If you’re in an environment where EVERYONE is cruel, niceness isn’t a good strategy. However, only a few nice people are necessary, even if majority of players aren’t nice

Surprisingly, rationality or morality of your fellow players is not a requirement for the Golden Rule to be an optimal strategy. Where your fellow players are irrational or wicked, overtime in a repeated game with the ability to see people’s history, being nice to others is the best strategy.

Some examples of repeated games in real life are trade and nuclear warfare. In both instances, if only one player decides to be cruel he gains an advantage over the other player, but ultimately, the best strategy is co-operation.

In repeated games where there are race to the bottom, mutually assured destruction or synergy effects, it becomes an even better strategy to cooperate with other players as the costs of cruelty and the benefits or co-operation are greater than usual.

You can read Axelrod’s paper here.