How Should We Make Decisions?
At some point, most of us have made a decision by flipping a coin. Maybe we were flipping for who would choose a location for lunch or perhaps we were flipping for higher stakes. Either way, we probably crossed our fingers or prayed to our lucky starts or did whatever we do for luck and hoped the coin would fall our way. The beauty of the coin flip is its simplicity and equal probability distribution. Because we can neither logically expect to win nor lose (assuming the coin is fair), by subjecting a decision to a coin flip, we remove angst from a decision because we know it has zero expected value - we are leaving the decision entirely to luck.
What if we could apply similar principles to other decisions we make? No, this is not a suggestion that you should carry a coin in your pocket to make your decisions, but you should apply the same expected value analysis. These principles involve only basic math and should be accessible to everyone. First, an example using the coin to prove its expected value will always be zero. Assume I bet you $1 on a coin flip (using a fair coin with a perfect 50/50 probability). How do I know the expected value is zero? The expected value is calculated as the payoff times the probability of the payoff occurring plus the loss times the probability of the loss occurring.
(Payoff)*(Probability of Payoff Occurring) + (Loss)*(Probability of Loss Occurring) = Expected Value
In this simple example, my payoff is $1, and that payoff has a 50% probability of occurring. My loss is also $1, and it also has a 50% probability of occurring. Inputting these variables, the formula looks like ($1)*(50%) + (-$1)*(50%) (note the negative number to denote the fact that it is a loss). This works out to $0.50 + -$0.50, which of course, is $0. If I am a mathematically rational, I'm indifferent about taking this bet because I am not expected to win or lose money. Using our formula, we know before the coin is even flipped what the range of outcomes is and what my expected value from the flip is. This framework may not offer groundbreaking insight into your next coin flip bet, but it can be applied to a much broader set of decisions. The limiting factor regarding how many types of decisions the value added formula can be applied to is only those decisions where the probabilities cannot be gauged with any reasonable degree of certainty. You will encounter these decisions, but for the vast majority of decisions, the value added formula will remove the angst from your decisions because you will already know what to expect from their outcome.
Let's look at a more complex example involving a decision we all make - investing. Some people say Bitcoin is risky and hackable and has absolutely no value. Some say it should be worth $10 million per coin. Assume you're intrigued by Bitcoin, and you do some of your own research and find some merit to both positions. You're torn because part of you wants to invest in an asset priced at $20,000 when it may rise substantially in value, but the other part of you can't pull the trigger on the purchase because you're afraid of seeing your $20,000 investment become worthless. Rather than dally in paralysis, you decide to use your Bitcoin research to assign a series of probabilities to Bitcoin attaining price levels you think are reasonable. Importantly, we must decide on a timeframe within which to assess the probabilities because an unlimited timeframe introduces too much uncertainty. A Bitcoin may be worth $100 million in 200 years (assuming it still exists), but we won't be around for that. Therefore, we must limit our probabilities to a defined timeframe.
In this hypothetical, let's assume that over the next three years, you think there is a 50% probability Bitcoin is worthless, a 25% probability Bitcoin appreciates to $35,000, a 20% probability Bitcoin appreciates to $50,000, a 3% probability Bitcoin appreciates to $75,000, a 1% probability Bitcoin appreciates to $100,000, a 0.5% probability Bitcoin appreciates to $250,000, and a 0.5% probability Bitcoin appreciates to $500,000. In our formula, this looks like this:
(50%)*(-$20,000) + (25%)*($35,000-$20,000) + (20%)*($50,000-$20,000) + (3%)*($75,000-$20,000) + (1%)*($100,000-$20,000) + (0.5%)*($250,000-$20,000) + (0.5%)*($500,000-$20,000) = expected value
Applying the expected value analysis, we see that based on our research and intuition, we reasonably think buying a Bitcoin will yield a return of $5,750. There are a few things we can take away from this result. First and most obviously, buying a Bitcoin for $20,000 when we expect its value to be $25,750 (which is the expected value plus the price we invested) is rational because it generates $5,750 of value. The more value a decision generates, the less we need to double and triple check our assessment of probabilities because a significant expected value leaves us with a margin for error, such that we can be a reasonable amount wrong in our assessment of probabilities and still generate positive net value. If, instead, I thought Bitcoin had a 10% probability of appreciating to $1,000,000 and a 90% probability of being worth nothing, I would have an expected value of $80,000, which is four times as much as the investment. Therefore, we feel safer making this bet because of the magnitude of the positive value.
Conversely, the lower the expected value, the more confident we need to be in our probabilities before acting because we don't have much margin of error. If I misjudge the probabilities by even a minor amount, the net value of my decision could be negative. A $5,750 net value on a $20,000 investment leaves me with about a 25% margin for error, which is not a lot when assessing probabilities in a field we aren't an expert in. Hence, it would be wise to gut-check the probabilities assigned before making this investment.
The biggest takeaway from this example isn't that we now have another way to prove our rationality but that we can remove anxiety from our decisions. If we don't employ the expected value formula, we are tied to the price we bought the Bitcoin for. Depending on your level of neuroticism, you may check Bitcoin's market price weekly, daily, or even hourly, each time hoping that the price has moved favorably or at least not unfavorably. You may wonder when to sell your Bitcoin and when to buy more, likely ending up in the euphoria or glum of an intensely volatile asset. But you don't have to. Not if we apply the expected value analysis to our decision.
If we expect the value of a Bitcoin over our timeframe to be $25,750, then we don't have to subject ourselves to the angst of price movements. Instead, we don't care if the price falls as long as the fall wasn't caused by a factor that impacts our probability assessment. And if the price rises, we know when to sell, again, as long as the rise wasn't caused by a factor that impacts our probability assessment. We should monitor our investment for news that changes our probability assessment, but if our probability assessment was reasonably thorough in the first place, only a rare "black swan" should alter your assessed probabilities over the given timeframe.
Further, we can make decisions with greater confidence. Rather than simply hoping for the best or following the market, or reading investment tarot cards, we assess the probabilities we developed through analysis and either accept them and make the decision to move forward or reject them and make the decision to say no. Going back to the example above, we assessed a 50% probability that Bitcoin would be worthless. Therefore, if we decide to buy a Bitcoin, we accept that probability. That means we aren't grinding our teeth in the next downturn or due to another CNBC article proclaiming Bitcoin to be a worthless asset - we've already accepted a high probability that Bitcoin may be worthless and bought it anyway.
This leads to the most satisfying part of employing the value added analysis in our decisions - it lets us tune out everyone else. If we put in the time to analyze a decision and assess the probabilities of potential outcomes from the decision, then we don't care whether others agree or disagree with us. We've done our homework and believe in ourselves and our judgment. That doesn't mean that we will always be right, but if we put in the time to do the analysis properly, we will find that more often than not we are right, and the outcome lands squarely within our range of assessed outcomes.
Given the formulaic nature of the expected value analysis, it is most easily applied to quantitative decisions but can also be used for qualitative decisions. For example, if you are considering whether to move, you might assign probabilities to happiness levels instead of using the analysis to determine a monetary value. Your analysis won't provide a precise result, but it will provide direction in addition to removing anxiety, enhancing confidence, and permitting you to tune out others.
A new way of thinking leads to a new way of living. By employing value added analysis in our decision-making process, we dampen uncertainty and free our minds from noise and anxiety. Now, it doesn't matter whether we are flipping a coin for lunch or evaluating a significant investment, we can't know the outcome, but we can know what to expect. And as long as we know what to expect, we shrink the unknown and, correspondingly, our fear of it.