A Thin Line between Chance and Responsibility

Let’s play a game. I flip a fair coin behind my back, and you guess “heads” or “tails”. After you guess, I reveal the result of the flip, and you win if you guessed correctly. Obviously, assuming no funny business, you have a 50% chance of winning regardless of your choice.

Simple enough, right? Let’s change the game slightly. This time, I flip two fair coins–one in each hand–and you guess “heads” or “tails”. If you guess “heads”, I reveal the coin in my right hand and keep the one in my left hand hidden. Otherwise, if you guess “tails”, I’ll reveal the left coin while concealing the coin in my right hand. You win if the coin that I reveal matches your guess.

Let’s denote these two versions of the game as V1 and V2 respectively. It’s not so hard to see that in this second case with V2, your chance of winning is still 50%. If you guess “heads”, there is a 50% chance that the coin in my right hand landed on heads. If you guess “tails”, there is a 50% chance that the coin in my left hand landed on tails. In fact, from your perspective, V2 of this game might as well have been identical to V1. Whether I flipped one or two coins behind my back had no impact on the outcome of the game. After all, you only get to see one of the coins, and both coins function identically. Is there even a difference between these two scenarios? Perhaps philosophically?

To answer that question in a more interesting manner, let’s say you have a time machine. Suppose we played V1 and you bet your life savings. You guessed “tails”, but after I revealed the coin, it turned out to be “heads”. Now you’ve lost your entire life savings. However, by seeing that the coin was revealed to be “heads”, you’ve learned some helpful information. Notably, you know that if you had guessed “heads” instead, you would have won. There’s no chance or doubt involved once you see the result. Since I flipped the coin before your guess, it’s guaranteed that the coin would have revealed to be “heads” irrespective of your guess. Fortunately, with your convenient time machine, you can travel back to the point when you made your guess and switch it to earn back double your life savings.

What if we had played V2 instead? Suppose you guessed “tails”, and the outcome was “heads”, so you lost your life savings. Would you have fared better if you had guessed “heads” instead? It turns out, by the simple fact that I flipped two coins instead of one, knowing that the result was “heads” doesn’t tell you anything about what would have happened if you had guessed “heads” instead of “tails”. You know that the left coin flipped to “heads”, but that doesn’t tell you anything about the coin in my right hand, which is the coin I would have revealed if you had guessed “heads” instead. Both were flipped independently and don’t affect each other. Sure, you might as well use your time machine to go back and change your guess to “heads” anyway, but your chances of winning are still 50%. Your chances of winning are no better than just playing again without using the time machine.

Okay that’s great, there is a difference established here. Given evidence of an actual outcome (the first guess), we can make inferences about the hypothetical outcome of a different action (the second guess after using the time machine). This inference is different depending on which version of the game we play. In V1, knowing the outcome of the first guess tells us exactly the outcome of the second guess. In V2, knowing the outcome of the first guess tells us no additional information about the outcome of the second guess. Still, time machines don’t exist, and you wouldn’t even have known which version of the game we were playing just by looking at the coin result if I didn’t tell you. So why is this even interesting to consider?

In terms of practical applications, it seems like you could never predict a hypothetical scenario that didn’t actually occur if you don’t understand the underlying rules of nature. For example, knowing the outcome of the coin flip wouldn’t have helped you predict the outcome under a different guess if you didn’t know which version of the game you were playing. And since time machines don’t exist, you can’t keep going back in time to verify it experimentally.

This ambiguity is, in fact, what makes this kind of phenomenon interesting. This is precisely the issue with trying to understand someone’s intent. Suppose Susie and Billy are walking together, and they spot a lost wallet on the ground. Billy looks in the wallet and finds a thick sum of money in it. Susie tells Billy to turn the wallet into the police, and Billy responds, “Of course I will, I would never steal the money.” They turn it into the police, and Susie is impressed by Billy’s honest nature. Still, despite the outcome, was that really Billy’s true intentions? Had Susie not been there, would Billy have kept the money for himself? Maybe Billy was telling the truth that he would never steal the money, or maybe he lied to conceal his true nature. At the end of the day, the wallet was returned, but the answers to these questions determine Billy’s honesty. Our impression of Billy’s character depends on the alternate realities where Billy finds the wallet under a different setting. We could continue to observe Billy in the future, but we can reserve judgment on him in this situation because we would never know what he was really thinking in the moment.

This kind of plausible deniability is often the issue that courts face when establishing blame and responsibility. Did John commit the crime? Was it an accident? Even if John committed the crime intentionally, did he do so under the order of his boss? Would John still have committed the crime if his boss didn’t give the order? Did John’s boss give him the order, not expecting him to commit a crime? These questions are impossible to answer definitively since we could never look into John’s or his boss’ head. At best, we can only infer based on collected evidence and assumptions.

In the context of the coin flip game, there are implications of responsibility as well. In V2, you may have lost after guessing “tails”, but who knows if you would have won if you had guessed “heads”? In V1, you absolutely know you should have chosen “heads” instead. In some sense, that makes it your fault for losing in that scenario. Interestingly, this implication of blame depends on which version of the game we play, which is indistinguishable from your perspective as the player.

Broadly speaking, the study of this kind of phenomenon falls under the umbrella of explainability and fairness. Perhaps disappointingly, the coin flip game shows that it’s generally impossible to inquire about a hypothetical outcome from realized observations alone (i.e. in the case where the player is unsure which version they are playing). Yet these questions about hypothetical outcomes are often the most important when it comes to understanding why something occurred or who is responsible for making it occur. Making educated guesses to these questions is tricky, yet it is how society progresses.

For a more technical understanding of this phenomenon, consult sources on causal and counterfactual explainability and fairness such as this textbook.