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Be Optimistic about People — Learnings from Game Theory

Game theory offers great insights — but only after adding miscommunication, we can apply the findings to humans.

The prisoner’s dilemma is one of the most common examples of game theory. It is simplistic — yet gives profound insights into the usefulness of cooperation. It goes something like this:

Imagine two prisoners, Maria and you, are accused of having done the same crime together. They are held in separate cells without any means of communicating. The police do not have enough evidence yet to incriminate both of you, but this is not known to the prisoners. So, the police officer in charge of interrogation proposes to both of them individually the following deal:

• If both confess, each of you serves two years in prison
• If one confesses, but the other one remains silent, the first will be set free, while the one who remained silent will serve 3 years in prison
• If you both remain silent, both of you will serve only one year in prison

There is no way that any of the prisoners can punish the other one after the sentence. Additionally, their reputation will not suffer, hence this game is called ‘unrepeated’ — there are no learnings for future games.

We can visualize the payoff matrix as follows.

What would you do?

Interestingly, given the unrepeated nature of the game, confessing is the best option. Why? Let’s just look at your payoffs:

• If Maria confesses, then it is best to also confess, otherwise, you will end up in jail for 3 years
• If Maria remains silent, then it is also best to confess, because then you walk free

If you make a survey though, then people are much more cooperative, than this thought experiment suggests. Why is that?

Repeated games

What part of this thought experiment is different in reality? It’s the last sentence in the description above, the fact that it is unrepeated. What happens if we change it?

There are two main ways to make this happen:

1. concrete number of rounds say 4 rounds
2. a random number of rounds (by introducing a likelihood of e.g. 10% that the game ends in the current round)

If we were to play 4 rounds of the above game, then you can react to Maria’s behavior in the subsequent round. And she knows that. So, would that make her remain silent? It would make sense because then you could both go to jail for just one year in the first round. And the second, and the third. But what about the fourth? Well, the last round of a repeated game is not repeated. So, the purely rational thing to do is, to confess (basically betray the other one).

Since the last round is essentially an unrepeated game, you already have to confess — and Maria will likely do the same. If she does it in the last round, then she cannot retaliate for your behavior in the second-to-last round. The result is that the best strategy is to always confess/betray.

So, why is this still so different to what most humans would actually do? Because there is a set number of rounds. If you change this little factor to unknown, everything changes and cooperation can be sustained for the entire time. Unknown means, both of you know that there will be multiple rounds, but you don’t know how many.

Here are some strategies to use:

• always betray
• always cooperate
• tit for tat (start by cooperating, then do whatever the other person did in the previous round)
• tit for tat with initial betray (same as tit for tat, but starts by betraying in the first round)
• tit for two tats (same as tit for tat, but switches behavior only if the other player repeats their behavior)
• grim trigger (start by cooperating, switch to betraying as soon as the other person betrays once)
• random (well, just randomly choosing every round)

It is interesting to notice the different environments required to make these strategies work. Always betray, always cooperate, and random always work. Tit for tats (both starting configurations) and grim trigger need the memory of the previous round. Tit for two tats requires memory of the last two rounds to work.

Now it is really interesting to see how different strategies do with each other. I have built an ‘arena’ in python. It allowed me to see what happens when differently sized amounts of players of different strategies are in one game. For multiple players, you can tweak the game to be about money as such:

In this virtual arena, people start with 0$ and you check how much money they have after a random number of transactions. Each round two players out of the player population in the arena are randomly assigned to enter a game with the previously defined rules. More advanced games might kick the e.g. 10% players with the lowest amount of money while duplicating the 10% with the most money. Then things quickly get into simulating evolution.

Who would do best? Three learnings:

1. More advanced strategies do better (of course, otherwise they would not exist)
2. The success of one strategy depends on who the other players are. Always confess is great with tit for tats, for example. Tit for tat with initial betrayal does not work well with tit for tat (they turn into always betray after the first round)
3. Overall, tit for tat seems like one of the best strategies

Application to human relationships

What is the learning here? Tit for tat is good? If you remember the title, this cannot be the whole story.

Here is my little tweak to the learnings above: introduce a little miscommunication!

Miscommunication technically works like this: every round the player’s choice of action has an e.g. 10% chance of being inverted (given that there are only 2 options, if it’s not the original intent, it must be the other one).

How does this change things? Proportionally, to how high the miscommunication strategy is, more forgiving strategies make more sense.

Miscommunication ⬆️ = more complex, more forgiving strategies work

I am not sure about how game duration factors into this, but I assume that longer games (those with a lower likelihood of being ended this round) make the above effect more pronounced because the likelihood that miscommunication actually occurs increases. If a game ends in a given round with a 30% likelihood, then it is rather short compared to a game that ends with a 0.0001% likelihood in a given round.

Anyways, back to miscommunication:

• 0% likelihood (see above, tits for tats is usually the best)
• 0% to <50% miscommunication likelihood: As the likelihood increases more forgiving strategies become more successful. The degree to which the other player’s move cannot be understood by complexity (which is always beneficial), needs to be mitigated with forgiveness or optimism. At a 10% miscommunication likelihood, tit for two tats clearly outperforms tit for tat while at 0% miscommunication likelihood tit for two tats is worse e.g. against always defect.
• 50% miscommunication likelihood: peak randomness, this is the setting where it is hardest to make sense of the other player
• 50%+ miscommunication likelihood: same as for below 50%, except all strategies, slowly invert (at 100% miscommunication always betray becomes always cooperate)

The above is especially pronounced if we turn evolution settings on (i.e. bottom 10% of players in a multiplayer game die while top 10% duplicate). In this scenario, a key ingredient for the success of one strategy after n rounds is their ability to deal with other players of the same strategy. Tit for tat, the most successful strategy in the classic settings is horrible when introducing miscommunication.

The above described gets further pronounced if we tweak the miscommunication feature to only invert cooperation actions. I think this is an accurate condition, as in human interactions miscommunication mostly is a good intention perceived badly, rarely the other way around. In this case, tit for tats is likely one of the worst strategies because even though they start out well, one will eventually miscommunicate, the other will retaliate and they will never leave this cycle, effectively driving themselves to extinction.

Application to humans

In my view, this last scenario is the closest to humans.

• Miscommunication happens
• Miscommunication tends to be about making a good intention seem bad, almost never the other way around
• We have a rather long game, it’s easy to conclude, that a human relationship can have millions of interactions over a lifetime

Hence, I argue, it is best to be tit for two tats or even more complex in how we forgive. The degree to how forgiving we are should correlate strongly with how much miscommunication we believe there is. If you read this and think you are likely a tit for tat player ask yourself: how much of what people do and say to you is miscommunicated (in the narrow sense above: meant to be cooperation, but is perceived/comes across as betrayal)? And if this is >0% then likely you should be more forgiving, even if it is slightly.

If your answer to reading this is ‘I am good at understanding humans’ you are claiming there is no miscommunication. And of course, miscommunication likelihoods are not equal for all people.

For logical completeness: The case with extremely high miscommunication is not applicable because at 50% one cannot speak of miscommunication and at >50% we would just redefine the original strategies.

If you are into these kinds of thoughts, you will probably enjoy this youtube channel which features animated game-theoretical analysis.