The (non)sense of emotional hedging

A ‘naïve’ normative perspective

Let’s start with what may sound like an absurdly general question for the topic at hand: why care about money in the first place? You may feel a rush of excitement from a stock in your portfolio surging or a prediction-market contract resolving in your favor, but you’re ultimately just looking at pixels on a screen. That money is only valuable insofar as you can later spend it on things you need or enjoy. If you knew it would be locked up in a bank account forever, that rush of excitement would seem, on reflection, to be misplaced.

Of course, you are allowed to spend your \$100 gambling winnings on nice dinners, massages, or necessities. But there’s nothing special about those particular dollars. You could just as well draw from the money in your savings or brokerage account, which you earned from working a normal job or what have you. Money is fungible, as the economists say. So, if you want to get a massage, a \$100 gambling win is only relevant in the broader context of your total wealth, which is hopefully at least an order of magnitude larger than that \$100. (If you have less than this, please don’t bet on sports!)

Suppose you have \$50,000 in a brokerage account when you witness your favorite sports team lose a championship — something that shouldn’t happen too often (except, perhaps, in Boston). You didn’t place any bets on the game, but spending \$100 at a favorite restaurant would be particularly helpful for your mental welfare at this moment. Is it worth paying this cost? Probably yes, so long as you don’t have any major expenses coming up. But whatever you answer, consider whether this answer would — or should — change if the amount in your bank account was \$50,100 instead because you just received an extra \$100 from your emotional hedge. Surely not! Although the \$100 difference in your total wealth isn’t meaningless, it shouldn’t impact your willingness to purchase a little treat for yourself when you’re having an unusually bad day. By holding tens of thousands of dollars in savings, you’ve already insured yourself against minor fluctuations in your mood. If money goes farther in bad times than good times, then you should plan to draw from your savings more during the rough times. A windfall \$100 doesn’t change that fact.

Now you might be thinking: “But it doesn’t feel that way! If I have to draw from my savings after suffering emotional pain, that makes my emotional pain even worse. Isn’t the whole point of emotional hedging to soften these negative feelings?” Hold on to that objection for a moment. First, let’s finish the standard economic explanation for why gambling is inadvisable.

From the analysis above, it may seem harmless to bet against your favorite team. If you get a rush from winning \$100 from gambling, what’s the harm? As a one-time endeavor for a relatively small amount of money, the harm is minimal. But, if you were to commit in the long term to this system as a strategy for emotional management, the harms could be significant.

Imagine that the online bookie or prediction market gives your team 1:1 odds, so a theoretical bet of \$100 against your team will either net you a \$100 profit (if your team loses) or a \$100 loss (if your team wins). Since these odds reflect the crowd wisdom of professional bettors, you have no reason to believe they are incorrect. If you could place this bet without incurring any further costs, you assume that it’s equally likely for your total wealth to change from \$49,900 or \$50,100. Your expected future wealth is, thus, just the average of these two numbers, i.e., the \$50,000 that’s currently in your account. Therefore, from this perspective, the bet is neutral. But there are a couple of issues. First, money has diminishing returns: the more of it you have, the less valuable it becomes for buying things like nice dinners or massages because you can already comfortably buy those things. Therefore, from the perspective of what money can get you (i.e., your utility) the average utility of a coin flip between \$49,900 versus \$50,100 is slightly less than the utility you currently have from \$50,000. Granted, when dealing with such a small amount of money, the role of diminishing returns should be negligible. The bigger issue with placing repeated bets against your team is the transaction costs. The bookies will take a substantial cut of your earnings, and prediction markets require a buyer or seller on the other side of your bet. Even if this costs you just a few dollars per transaction, the losses will add up over time, and you should expect to lose money from your gambling habit.

To summarize, from the traditional normative perspective of expected utility theory, emotional hedging seems like a foolish approach to managing emotions. If you’re having a bad day and could spend a little money to cheer yourself up, that option is already available to you; getting an extra \$100 shouldn’t meaningfully influence your willingness to splurge. By focusing on your gambling winnings as a special windfall to use for emotion management, you’re making the classic error of mental accounting — acting as if the source of this particular pool of money changes its usefulness. Moreover, in the long run, this is a costly error: ultimately, you can expect to have less total money in your bank account to hedge against future (non)emotional problems.¹

Perhaps you feel that this perspective on emotional hedging ignores key aspects of psychology, though. Our emotions aren’t always responsive to cold, economic analysis; for better or worse, we do get excited by gambling windfalls and attend to how we win or lose money. Even if we would be richer without these biases, the emotions are real and unavoidable. And aren’t the emotions precisely what we’re trying to manage when we emotionally hedge?

Taking emotions seriously

How are our emotions shaped by outcomes? Although it’s not a theory of emotion per se, Amos Tversky and Daniel Kahneman’s Prospect Theory offers a compelling, empirically supported account of how events get translated into subjective values for decision-making, summarized in the following figure (source):

In a nutshell, Prospect Theory is a psychological — decidedly not normative — theory that proposes that people’s preferences are shaped by changes in perceived states of the world, such as a \$100 win from gambling. Note the contrast to the discussion above, in which total wealth (rather than a change in wealth) was what I argued should matter. Prospect Theory further proposes that the experience of a loss (e.g., losing \$100) hurts more than its equivalent gain (e.g., winning \$100).

As we’ll see, the “reference point” around which states are perceived to change is not always easy to define, as it is in the eye of the beholder. It’s a psychological default, which could differ across people or situations. For the purposes of hedging the emotions of a sports outcome, the natural reference point is your emotional state before the winner is determined. Observing your team lose will shift your subjective value negative by moving you into the “losses” side of the plot, while observing your team win will have the opposite effect.

Bear in mind that the shift leftward on the x-axis from a sports loss may not be the same distance in outcome space as the shift rightward from a win. We can see this by mapping a sports win or loss into its hypothetical equivalent-feeling monetary gain or loss, respectively. For example, maybe witnessing the Patriots win the Super Bowl in 2012 would have felt equivalent to my winning \$300 (e.g., because I would get to celebrate at the parade, reminisce with my friends for years to come, and so on), whereas witnessing them lose only hurt as much as losing \$100, as I could still look forward to future years of dominance. Whatever the exact amounts, it’s useful to map a sports outcome (or anything else you’d want to hedge) to subjectively equivalent monetary changes for the sake of our discussion about emotional hedging, so we can put the focal event and the hedge in the same monetary outcome space.

Prospect Theory specifies how shifts from the reference point map on to subjective values via the function \(v(x)\). As can be seen in the plot above, this function is concave for gains and convex for losses. That is, each additional dollar of gain is associated with less of a change in positive emotion, and each additional dollar of loss is associated with less of a change in negative emotion. At the same time, the rate of change is steeper for losses than equivalent gains.

If we take Prospect Theory’s subjective value to represent emotional valence, we can calculate an “expected emotional state” in a similar way to how we’d calculate an expected utility, but using \(v(x)\) instead of a utility function. For example, to say that I ‘expected’ to have a positive emotional state after the 2012 Super Bowl implies that

\[ p*v(+\$300) + (1-p)*v(-\$100) > 0, \]

where \(p\) was the probability I estimated for the Patriots’ winning.² Given that the team was a big favorite to win the game (e.g., \(p = 2/3\)), this expected subjective value was clearly positive even without a hedge. (Presumably, this fact at least partially explained why I was so enthusiastic about football at the time. It would have been much harder to be a New York Jets fan.)

But just because I expected to be happy after the game doesn’t mean I couldn’t have expected to be even happier with a hedge. Here’s one way to look at it. If a Patriots loss felt like losing \$100, I could have taken a bet at the prevailing 2:1 odds that costed me \$50 if the Patriots won, but netted me \$100 (or slightly less with transaction costs) if they lost. When the Patriots lost, I would no longer be ‘losing’ \$100. In a sense, I would lose nothing because the profit from the hedge cancelled out the \$100 worth of pain I would have experienced without the hedge. In the likelier event that the Patriots won, I would only feel like I was winning \$250 because of the \$50 I burned. So my expected emotional state would simply equal \(2/3*v(+\$250)\). If my emotions conform to the Prospect Theory curve, this is a happier expected state than the expectation without the hedge, which we calculated earlier. The difference in negative emotional experience between \(v(-\$100)\) and neutrality (i.e., \(v(\$0)\)) is large because of the steepness of the loss curve and its convexity; meanwhile, the loss in positive experience between \(v(+\$300)\) and \(v(+\$250)\) is much smaller because of the shallowness of gains and concavity of the gain curve. A big win for hedging!

But wait. Is that how emotions work? If I had taken this bet, I would really feel nothing if the Patriots lost, as if two outcomes — the Super Bowl loss and the profit from the bet — merged into a unitary experience? Having gone through this process several times now (albeit not with the intention of completely hedging my negative emotion), this has not been my experience. Instead, I feel initial disappointment from the loss (or joy from the win), followed by some satisfaction from winning the bet (or annoyance from spending money). I experience two emotional events, not one.

This is a fatal problem for the hedging strategy. On this two-experience model, Prospect Theory would imply that the 2012 Patriots loss with the hedge would lead to an initial feeling of loss, \(v(-\$100)\), and then a subsequent feeling of gain, \(v(+\$100)\). And because losses loom larger than gains,

\[ v(-\$100) + v(+\$100) < 0. \]

That is, the hedge doesn’t completely cancel out the pain from the Patriots’ loss on the scale of emotional valence. Even worse, if the Patriots had won, the \$50 price of the bet would be experienced as a distinct monetary loss, \(v(-\$50)\), subtracted from the happiness of winning the game, \(v(+\$300)\). This hurts much more than a \$50 reduction of a gain.

In general, from this perspective, the bet is not softening an emotional loss at the expense of slightly reducing an emotional gain; it’s addinga new monetary gain at the expense of introducing a new monetary loss. Because the reference point resets, Prospect Theory says that this is an undesirable tradeoff. Losses loom larger than gains.

Perhaps you’re skeptical of my phenomenology or believe that your emotional system works diferently. I don’t mean to rule out that possibility. But keep in mind that the reference point from Prospect Theory, which has been studied to death, is extremely fickle and sensitive to minute changes in framing. Tversky and Kahneman demonstrated that most people respond differently to an opportunity phrased as “You’re given \$2000 and lose \$1000 if a coin lands tails” and one phrased as “You’re given \$1000 and win another \$1000 if a coin lands heads.” It doesn’t seem like it would require much cognitive work for the emotional system to parse each of these options as “50% chance to win \$1000, and 50% chance to win \$2000.” But, to the average person, the options feel different! The emotional system apparently isn’t doing the simple math of combining the two parts of these statements. And if even a subtle change of wording for two in-kind monetary transfers doesn’t induce an integrated emotional state, what hope is there for combining the complex pain or joy of a sports oucome with the pleasure or pain from a monetary bet?

In short, even if we treat Prospect Theory as an unavoidable determiner of our emotional states that needs to be hedged on its own terms, the logic of hedging these emotions is far from straightforward. It rests on a dubious presumption that life’s everyday pains and pleasures smoothly integrate with the experience of winning or losing money.

To be sure, people’s emotional systems vary, and not everyone shows the classic decision patterns of Prospect Theory to begin with. So you may have an unusual emotional system for which hedging works well. But, for a typical person, hedging is a questionable approach.

Don’t be a slave to irrational emotion

Notwithstanding these issues about how we emotionally integrate gains and losses, we ought to be asking a more fundamental question: are we really as beholden to our emotions as the above perspective suggests? If Prospect Theory is a fixed feature of our psychology, like any other feature of our environment, then we inevitably pay an irrationality tax by emotionally hedging. And when we have less money because we’re chasing the short-term emotional highs of gambling windfalls, we can do fewer pleasurable things with our money in the long run. What if, by recognizing this absurdity, we can retrain our emotional system?

Clearly, we don’t have unlimited power to shape feelings. I recognize that I would be healthier and need to spend less time exercising if I didn’t enjoy eating chocolate. Given that my appetite functions in this way, I don’t regret indulging in dessert. But if I could redirect some of the pleasure I get from chocolate to a healthier food, like broccoli, I would surely take that trade. I can’t.

I don’t think this is the correct metaphor for Prospect Theory, though. By recognizing the flaws in our intuitive, “System 1” approach to decision-making by learning behavioral economics, we can debias ourselves. For example, imagine you’re buying a new car that costs around \$20,000. The dealership near your house offers the exact same car for \$500 more than a dealership out of state, which would take you a couple of hours to get to. Is it worth it for you to take the long trip to save some money? There is no one-size-fits-all answer to this question; it depends on how rich you are, how inconvenient it would be for you to take the time to get to the out-of-state dealership, and so on. But if you rely just on your default emotional sense about whether the trek is worthwhile, you can easily be misled. Prospect Theory implies that the \$500 savings will feel much more worthwhile if you were making the trip to save that much on a television that costs \$600 at baseline. It also implies that the savings would feel less worthwhile if you were thinking about a \$500 swing in a stock portfolio of hundreds of thousands of dollars. Should you just accept that this is the way your emotional system works even though it’s inconsistent with the fact that money is fungible? Of course not. You should run thought experiments like these in your head: How would you feel if the car were a smaller, or larger, purchase? What would you do with an extra \$500? Etc. Over time, you may come to feel differently than you initially did.

We can ultimately take a similar approach to emotional hedging, even if the target of hedging is the emotional system itself. To start, perhaps we can reflect on our reasons for caring about our favorite sports teams (or whatever we’re emotionally dealing with) in the first place. Is an extreme negative reaction to a team’s loss an inevitable consequence of being a fan, or can we reframe the loss as less of a big deal to our long-term welfare? Maybe we can’t, or don’t want to because it seems disloyal. Well, then reframe the hedge. Is a possible gambling windfall with transaction costs the best approach to managing our potential disappointment? We should only care about money insofar as it buys us stuff. So a change in our total wealth that we might regard as basically noise in the context of our stock portfolio shouldn’t feel like anything really. And by imagining the gambling winfall as part of this broader context, we should begin to feel better about occasionally dipping into our savings to treat ourselves on a hard day.

I realize that this is all easier said than done. Before writing this post, I placed some bets against the underdog Patriots’ winning the Super Bowl this Sunday. On reflection, this made little sense. I expect the Patriots to lose and therefore don’t even have serious negative emotion to hedge. Am I just upset about all the money I’ve lost betting against them this year, even though I know it’s foolish to chase past losses in hopes of winning it all back? I’ve let my base emotions get the best of me. Perhaps it’s time for me to take my own advice.