People are asking me on Twitter what my ‘core thesis’ is on Covid-19. I don’t really have one, but the closet thing to a core thesis I have involves risk-benefit analysis and decision theory. A little explanation of this would be in order first. (If you already know this sort of thing, feel free to skip it, it’s basic. I’ll draw some lessons from it in a later post. Or read it and make suggestions about how it could be improved.)
I’m going to greatly simplify matters in order to make this easy to understand. (These simplifications may offend some, but I find those people are often the ones who aren’t very good at getting this sort of stuff across to newbies.)
Utility
The first concept we need to introduce is ‘utility’. This is just how good some state of affairs is for you, or for whoever we are concerned with (your family, a group, a country, whatever).
What do we mean by ‘good’ here, you may ask. At the moment it doesn’t really matter. It could be an entirely subjective idea of how much you value some state of affairs. The general principles aren’t really affected by exactly how we understand ‘good’ here.
We can assign a numerical score to the utility to express how good it is. The better the state of affairs is, the higher the score. If something is bad, though, we can assign a negative score, and call it a ‘disutility’.
Again, you’re going to interject and say ‘But on what basis or principles do we assign scores? And what’s the scale? And isn’t this entirely subjective’? These are all fair questions. There are two things I can say at this stage. One is that the basic principles of decision theory can be understood without worrying too much about these issues now. However, decision theorists, especially ones concerned with public policy, do largely get around these issues by using monetary values for their utility scores. This does not get around the problem that not everyone values money in the same way, but public policy types will just reply that they are sidestepping that by just concerning themselves with monetary matters only, and they’re doing their risk-benefit analyses on that basis, and leaving further discussions of the importance of money for others to discuss.
I’m going to use money in some of my explanations, mainly because it makes things a bit easier to understand.
In decision theory we are usually concerned with the utility of possible states of affairs in the future, and the rationality of choosing one action over another. This may sound a bit abstract, but we can get a grip on it by considering a familiar example: coin-tossing.
Coin-tossing
Suppose you are faced with a coin toss. You will win £100 if you call correctly. You lose nothing if you get it wrong.
Let’s use the monetary value as the utility, so that the utility of calling correctly is £100. But what’s the utility of calling, say, heads before the toss and before we know whether you were right? That is, what’s the utility of the action? We can’t just use a straight utility score of £100 here, because there’s a 50% chance that you’ll call incorrectly. Same with tails.
Expected utility
This is where we introduce the concept of ‘expected utility’. If you’ve ever done any economics you should be familiar with this concept. It’s just the utility of the outcomes weighted by — taking into account — the probability of the outcome.
So the action of calling heads has an expected utility of £100 multiplied by 1/2 (ie. 0.5, or 50%). This gives us £50. So our expected utility is £50. And the same applies for a call of tails, that act also has an expected utility of £50, because we have the same outcomes and probabilities.
Expected utility should be a concept that is easily understood by most people. Even if the later complications get hard to follow, the general idea shouldn’t be too hard. Basically, you have to take into account the probabilities involved when making an action, you can’t just consider the possible outcomes without weighting them by how likely they are. In fact, most people already understand this sort of idea even if they’ve never been formally introduced to the concept of expected utility.
Complications
There are, though, some complications already that we need to talk about.
First of all, we can assign expected utility scores to possible outcomes, and also to actions, and these will usually differ, because virtually every action has more than one possible outcome.
Firstly, we need to assign an expected utility score to each possible outcome of an action (including negative disutility scores for bad outcomes) by looking at each outcome’s utility and probability.
To get the expected utility score for an action, though (as opposed to an outcome), we have to add up the expected utility scores of each possible outcome of that action. Outcomes that have a negative score cause the number to get smaller, not bigger, and some actions will have negative expected utility scores once we have done the adding up.
We can see this even in our simple coin toss example. Calling heads actually has two possible outcomes: the coin comes up heads, or it comes up tails (each with a probability of 0.5). With the former, you win £100, with the latter you win nothing (and lose nothing). The expected utilities of the two possible outcomes are, therefore, £50 and £0 respectively, which added together give you a £50 expected utility for a call of heads. (This also applies to a call of tails.)
But let’s change the scenario a bit, and introduce penalties for an incorrect call. Suppose you will lose £10 if you call incorrectly. This means that the outcome of the coin coming up tails when you’ve called heads has an expected utility of -£5 (-£10 times 0.5). That means our overall expected utility for calling heads is £50 + -£5 = £45 (ie. adding the expected utility of the two possible outcomes together). As the same applies to a call of tails, it’s clearly a game worth playing, as your expected utility is positive to the tune of £45 regardless of whether you call heads or tails.
Different expected utilities in the coin toss
Now let’s make it more complicated. Let’s change it so that we have different expected utilities for heads and tails. Suppose if you call heads and you’re right, you win £100, but if you’re wrong you lose £40. If you call tails and you’re right you win £120 but you lose £80 if you’re wrong. We can put this into a table to help us keep track of it, and work out the call we should make (or if we should play the game at all):
——————————————————
Coin is heads (0.5) Coin is tails (0.5)
Call heads: £100 x 0.5 = £50e.u. -£40 x 0.5 = -£20e.u.
Call tails: -£80 x 0.5 = -£40e.u. £120 x 0.5 = £60e.u.
Expected utility of calling heads in this scenario: £50 + -£20 = £30
Expected utility of calling tails in this scenario: -£40 + -£60 = £20
——————————————————
Standard decision theory says that the rational action is the one with the highest expected utility. So it’s more rational to call heads in this situation than tails, because it has an expected utility of £30, whereas with tails it’s £20.
If you’ve followed this you’ve already understood the basics — all right, the very basics — of standard decision theory, and standard risk-benefit analysis.
The real world
In the real world, of course, things are much more difficult than these simplified examples. For instance, we can’t always be sure that we’ve really listed all the possible outcomes that may result from an action. Another difficulty is that we often don’t have a very good idea of how probable real-world outcomes are. Probability with coin tosses is easy, it’s much harder with anything that involves people and economic and social systems. The general principles do remain the same with more complex situations, but actually applying them in reality is very hard. Often they’re so hard that whatever you come up with using decision theory is not going to be any better than a hunch. Supposed experts may disagree hugely on these matters, so using this sort of decision theory doesn’t guarantee in the slightest that we will choose the more rational action.
Nevertheless, some general principles remain, and are relevant to Covid-19, as I will explain in a later post.
20 thoughts on “A risk-benefit analysis primer”
As always, the question is — Who is the target audience? What will speak to that audience? Is the concept of utility the best approach?
There are many problems with the concept of utility. One example — Bill Gates feels bored and wants to watch someone play Russian Roulette. Standard 6-shot revolver with one bullet.
Bill puts $100 on the table and invites people to play — no charge. Spin the barrel, point the gun at your head, pull the trigger, put the gun down, pick up the $100 and leave. Expected value $100 * 5/6 = $83. Sounds like taking money from a baby, but no-one wants to play Bill’s game.
OK, says Bill, and ups it to $1,000. When no-one still wants to play, he ups it to $1,000,000. How much money would Bill have to put on the table before you personally would be willing to play? Now we are taking “utility”.
Sometimes the downside outcome of a probability calculation is simply unacceptable. In any single play, we never get the Expected Value — we win or we lose. I guess that many of the people who are excited about C-19 fall into the “If we can save one life” camp, and don’t care about the cost to society as a whole.
The Bill Gates scenario doesn’t present any problem at all for decision theory or the concept of utility. The expected utility of playing Bill’s game is most definitely not $83! You have simply failed to take account of massive disutility of the possible outcome involving you being killed, which has a non-negligible probability of 1/6th.
It sounds like we are trying to say the same thing.
Each of us would in effect (and rightly) put infinite value on our own life, and thus the utility of playing Bill Gates’ game would always be negative. Bill could not put enough money on the table to give the game a positive utility.
In the case of Covid-19, several politicians have reportedly made “if we can save one life …” types of comments to justify the over-reaction — effectively saying that crashing the economy and making millions unemployed with all the associated consequences pales in comparison to the value of a single human life.
That reported attitude is why I wonder if utility will be a useful concept to get certain people thinking about the downsides of trying to save that single life.
You see this sort of thing on that Noel Edmonds show a lot:
Walk away and get 25k
At the next guess you have two offers of 80k and 1p. People will walk away with the 25k. They don’t want to take the risk of losing it all.
Then we have marginal utility – I’ve got two bikes worth 400 and 200. The second bike is worth a lot less than half the first one – it’s just nice to have a spare and the space for it. Not going to be gutted to the tune of 200 if it’s stolen. If both are stolen I will be gutted to the tune of 600.
I’m interested to see where you’re going with this, but it has to take into account how individuals perceive the value of stuff they can be guaranteed.
I think if politicians could pitch to the public that if we come out of lock-down tomorrow and the vulnerable 10 weeks later then we’re all guaranteed a holiday in the UK this year worth 250 units ( you might be able to go abroad anyway ). If we come out of lock down much later and risk is CV-19 spiking back up again in the autumn then options from 0 holidays anywhere at all to a foreign holiday worth 800 units are in play.
Offer the public something good that they can be sure they’ll have.
The way your average bod is going to see the Russian roulette thing in the context of Covid economic arguments is that you have the gun, you’re pointing it at his granny’s head, and you win money if you shoot granny.
Do people, in the west, actually die because of even a vast economic shutdown? We’re clearly wealthy enough and have sufficient social security setups to keep everyone fed and sheltered, if it ever came to that. There are hypothecated deaths – in my field the amount of drug research that is being shut down both directly and indirectly because of covid means products will be delayed to market, months to years, and in that time lives will be “lost” (not extended) as a result.
But is there direct, observable mortality arising from economic shutdown, in a wealthy society? That seems worth asking given that we seem to be all so risk-averse and unaccepting of our own mortality.
I’m surprised we haven’t heard the term QALY (Quality Adjusted life Years) used in the press recently, when assessing the response to Covid-19. I was under the impression NHS departments use this all the time when deciding which drugs and treatment provide value for money.
Im sure I’ve seen similar analysis used in other areas of public life too. I remember when I used to live on the same road as a school, the local council proposed some traffic calming measures when I looked up the application online it even mentioned that the current layout led to on average 1 death per year and it had done a cost benefit analysis on whether the cost of the works would be necessary.
If this is the case why is the government pretending that these kind of calculations don’t go on all the time. Even if we take the most extreme worst case scenario of the modelling of deaths caused by the virus the cost per QALY saved by a lockdown will be far in excess of the cost used by NHS and other departments in normal times.
“Bill could not put enough money on the table to give the game a positive utility.”
I bet if Bill put £1bn on the table he’d have plenty of takers for sure. Hell, that kid in the US got his girlfriend to shoot him with a Desert Eagle, with only a thick book as ‘armour’, just for the youtube lolz. And paid the ultimate price. There’d be takers from the poorer parts of the demographic from £100k upwards. Not least because death would not have the same negative utility for everyone. Got terminal cancer and a family to support after you’re gone? I’ll take that gun Bill, for £500k. How about people who are suicidal anyway? Then its a win/win, you either top yourself or win millions. Just because you wouldn’t take that bet for any sum doesn’t mean there isn’t someone who would. Not everyone values their life at infinity. If they did no one would ever charge a pillbox, or rescue his wounded mate from no-mans land.
Oh Yes Bloke. In the USA after the financial crisis 2008 suicide alone went up 4%. In the uk it was nearer 6% I think.
I don’t think it is even a contested outcome either – see
http://inproportion2.talkigy.com/fear_impact.html
http://inproportion2.talkigy.com/facing_catastrophe.html
http://inproportion2.talkigy.com/treatment_delays.html
The young mum struggling to get medical attention for a sick 13 week old is a case in point.
Even NHS England is concerned –
https://drmalcolmkendrick.org/2020/04/06/covid-with-of-or-because-of/
Dr Kendrick’s site is well worth a browse through – the pieces on Placebo, and Statins are alarming (and I am not easily alarmed)
Going back to the very start of this piece – your core thesis on COVID-19. Are you looking to make suggestions to what we could do now, or looking to examine the government’s decisions earlier?
Do people, in the west, actually die because of even a vast economic shutdown? We’re clearly wealthy enough and have sufficient social security setups to keep everyone fed and sheltered, if it ever came to that.
This seems so mind-bogglingly naïve coming from who I assume is the same BiG at Tim Worstal’s place. Really quite Guardianesque and Murphyish.
Oh Yes Mark – QALY
A couple of simplistic articles by Toby Young from early in the crisis.
https://thecritic.co.uk/has-the-government-over-reacted-to-the-coronavirus-crisis/
https://thecritic.co.uk/a-response-on-the-cost-of-the-coronavirus-shutdown/
Simplistic as I say, but they give a flavour of the problem you highlight.
Sorry, that came across as much more confrontational than I intended. Visiting too many blogs, writing too quickly.
Hector, having read your novel I didn’t expect to see you coming out as a utilitarian economist. I’d suggest that the novelist Hector Drummond has a better handle on human motives than the economic rationalist Hector Drummond who’s contradicting ‘him’.
Once basic needs are answered for utility / economic rationalism has negligible explanatory value. Just because we can rationalise in terms of economic utility from an objective or neutral / third person / ‘scientific’ standpoint it doesn’t follow that economic utility supplies the motive.
As Roger Scruton put it, economics stands to politics as neurology to friendship. We could give an exhaustive account of what friendship is physiologically but no form of body scan or configuration of neural activity could give a clue to what friendship is for the friends themselves which can only be understood subjectively – ‘subjective’ here meaning human subject not some private inner-realm or personal opinion.
Economics is at best an index of rivalry and status. As when various groups of workers went on strike in the 70s over “differentials”. Nothing to do with economics as such. The signalmen got an extra £5 a week so the drivers wanted £10. Because value for humans is social not individual.
A century ago a car would confer the status of a private jet or lambo today. Availability is at the expense of value which is based on scarcity. Abundance creates a market for asceticism. Hence appeal of Green politics to the affluent. And if I’m never going to be super-rich myself I can still advertise my value through self-denial. I can’t lose.
We need to distinguish between appetite which is individual and desire which is social. If I’m hungry I’ll eat what’s available to me. Otherwise desire is “mimetic” i.e. based on others. Even the marketing of food relies on ‘models’ i.e. people possessing some quality that I feel myself lacking, typically fame, good looks, cheerful friends, athletic prowess etc. We might think we ourselves are above such base suggestibility, our bank statements suggest otherwise.
In the novel Conquest’s Third Law is invoked and I’d suggest that’s the most economical explanation in the current scenario:
“The simplest way to explain the behavior of any bureaucratic organisation is to assume that it is controlled by a cabal of its enemies.”
In other words its *behaviour* will contradict its ostensible purpose. The bureaucratic organism exists at the expense of its base or host. That isn’t to impute ill will to any individuals. If officialdom are pursuing economic collapse it doesn’t follow that they’re motivated by malign intent. In their milieu they’re doing the right thing. Of course it doesn’t exclude it either.
The broader point of Conquest’s maxim is that civil society exists at the expense of the bureaucratic state, as opposed to the classical conception of state serving society, which is to say electorate and officialdom – LibLabCon being PR front – can be understood as political rivals.
According to the classical economic model officialdom should be held to account by a ‘free press’. But the media are controlled by a tiny number of global plutocrats.
We’re supposed to be against “socialism” because unaccountable economic power spells political despotism. Billionaire plutocrats whose wealth immunises them against the exigencies keeping the rest of us in check should be opposed on the very same grounds.
One can only speculate as to the motives of a Bill Gates. What we can safely conjecture is that he’s no less inclined to assert himself politically than the plod telling people to stay out of their gardens and so on, even if it’s at a somewhat more rarefied level of a global vaccine.
Even if Johnson and his Conservatives identify as “libertarian” or “conservative” nothing grants immunity to political power. Besides, if anyone had left England ten years ago and returned there’d be nothing to suggest any change to political colour of government.
Non-EU immigration numbers have grown under Conservatives. Extension of police activity and criminal justice system as agents of political enforcement in service of multicultuarlism, mass marketed as Diversity / ‘anti-racism’ has increased ‘exponentially’. One might say Brexit but Brexit was originally a Labour cause – members rather than leadership.
Judged purely by their deeds, which could always change subject to what Heidegger called the “peculiar dictatorship of the public realm”, there’s no reason to suppose that the endgame isn’t economic ruin precipitating nationalisation on a vast scale. Communist ends by Conservative means. Which is no more than to say that people are political, i.e desire power over others, which was Scruton’s argument against a politics in service of economic liberlalism.
The only thing protecting us, in his view, being our inherited institutional arrangements which themselves rely on a degree of shared loyalties.
Arguably an atomised populace with no loyalties greater than to their soaps/football is as congenial to officialdom as to the purveyors of bread and circuses. Hence de facto coalition between officialdom and global plutocrats.
It was a genuine question as to what the sources of these deaths are. I know about the suicide one, the indirect ones (smaller economy means less money for healthcare). I know there are others. I am looking for ammunition to the “costing money is better than costing lives” arguments from leftists.
The QALY is used by drug purchasers to maximise the utility of limited resources. If the resources increased, they would happily buy more products with fewer QALYS per unit cost.
It doesn’t work the other way around – that we are happy to sacrifice some number of QUALYS to make more money.
I’d just like to say absolutely top marks. Great post.
Hector, Thank you for the excellent, clear analysis on this blog. But care must be taken in discussing money (= life chances) not to fall into a political/emotional trap that might be used by those who wish to denigrate your spotlight.
Dr John Lee put it well: “The moral debate is not lives vs money. It is lives vs lives. It will take months, perhaps years, if ever, before we can assess the wider implications of what we are doing. The damage to children’s education, the excess suicides, the increase in mental health problems, the taking away of resources from other health problems that we were dealing with effectively. Those who need medical help now but won’t seek it, or might not be offered it. And what about the effects on food production and global commerce, that will have unquantifiable consequences for people of all ages, perhaps especially in developing economies?”
I’m not a utilitarian economist, or any sort of utilitarian, and I certainly don’t advocate generally using this sort of population-level risk-benefit analysis to make decisions. But this is basically what the government is doing, even if they’re not explicit about it. I’m just pointing out that if we use standard decision theory here it seems more likely that no lockdown wins out over lockdown, once we take into account the low probability of Covid-19 being a mass killer.
Yes I *know* you’re not: I’ve read your novel. I should have qualified that initial remark but didn’t. Even if the utility argument is bogus, in the sense of politically motivated rather than merely false, that’s not a reason for not rebutting it on its own terms with truth and facts, as you’re doing. I hope I’ve made that clear in other comments. If anything your facts and figures have all the more force for being put dispassionately, unsullied by your no doubt deplorable political tendencies.
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