The unbearable lightness of expected utility

In chapter 8 of my book, Risk Intelligence, I discuss the concept of expected utility.  To calculate the expected utility of a course of action, the first step is to estimate – separately – the probability of success, and the potential gains and losses that success and failure would entail.  Next, one does a little math, multiplying the probability of success by the potential gains, and multiplying the probability of failure by the potential losses.  Finally, we adding these two figures together to end up with the expected utility of that course of action.  After doing this for each possible action, the rational choice is to pick the action with the highest expected utility.

Expected utility is an abstract concept. It doesn’t refer to any actual win or loss: it’s the average amount you would win or lose per bet if you placed the same bet an infinite number of times. But this ethereal figure takes on an almost physical nature for the expert gambler, looming even larger in his consciousness than the actual profit or loss that hypnotises the rest of us. Expert gamblers see something different when they look at a poker table or a roulette wheel. Most people see a range of prizes; they see a single abstract figure.

This was brought home to me one night when a highly successful gambler invited me to accompany him to a casino. This particular gambler made all his money betting on horses. He shunned casinos, and only ventured in on this occasion because he wanted to give me and his other companions the thrill of seeing someone playing for high stakes. We followed him over to the craps table.

Craps is a dice game in which players place wagers on the outcome of the roll of two dice. My gambler friend proceeded to show off by placing bets of 1000 US dollars on the pass line, one after the other. A pass line bet is won immediately if the first roll is a 7 or 11. If the first roll is a 2, 3 or 12, the bet loses (this is known as “crapping out”). If the roll is any other value, it establishes a point; if that point is rolled again before a seven, the bet wins. If, with a point established, a seven is rolled before the point is re-rolled, the bet loses (“seven out”). A pass line win pays even money; in other words, my friend stood to win or lose a thousand dollars on each bet.

However, as he told me later over cocktails in the casino bar, it wasn’t this figure that was at the forefront of his mind. Instead, he just treated each bet as a bit of fun that cost him fourteen bucks. Although not a casino gambler, he was familiar enough with craps to know that the expected value of a pass line bet is -0.014. And this meant that, on average, each bet of a thousand dollars would leave him fourteen dollars less well off. The actual profit or loss at the end of an hour on the craps table could be anything from plus fifty thousand dollars to minus fifty thousand, and my friend would be aware of this figure too. But the figure that mattered most for him was the expected value of minus fourteen bucks per bet. That’s not a good-value bet, of course. Which is why, in any other circumstance, my gambler friend would never play craps.

Scenario planning and probabilities

First, a caveat; I don’t know much about scenario planning, so the following comments may come across as rather simplistic to those well versed in this area.  Also, it is probably rather presumptuous to be so critical of something I know so little about. So consider this post as an opening gambit rather than a considered conclusion.

I recently exchanged a few emails with a guy who does scenario planning for a non-profit organization.  When I asked him if he got people to attach numerical probabilities to each scenario, he replied: “We don’t do probabilities, but instead run workshops and interviews to get a sense of where people’s mental models are in terms of how things might turn out.” The problem with this is that weasel word, “might,” which could mean anything from “extremely unlikely” to “almost certain”.

For example, suppose the folks at the Pentagon are mapping out possible scenarios that might follow a US invasion of Syria, such as:

  • Invasion is successful with minimal human and financial cost, Syrians welcome the troops and quickly set up a prosperous, democratic and liberal society that becomes a strong US ally and force for positive change in the Islamic world.
  • Invasion is a complete disaster with massive cost and casualties, resulting in a devastated Syria split into violent fiefdoms, including one run by Assad and another by Al Qaeda. The US is humiliated both militarily and by revelation of major scandals and atrocities. Many US troops are prisoners.

Plus various intermediate possibilities.

So far, so good.  The precise details in each scenario are not that important, since the scenarios are really just placeholders for a set of outcomes arranged in order of preference.  The real problems begin when we go from scenarios to decisions. For unless we have some idea of how likely each scenario is, it will be impossible to assess the expected utility of various mitigating strategies.  There may be no point in spending billions of dollars to avert a worst-case scenario if the probability of that scenario occurring is very low.

The Intergovernmental Panel on Climate Change (IPCC) attaches numerical probabilities to various scenarios it discusses in its reports. Everyone else who does scenario planning should do the same.

The expected utility of drone strikes


A recent article in The Economist asks whether US drone strikes in Pakistan help or hinder the war on terror. A friend of mine explained to me why he thought they did more harm than good. Suppose a drone strike takes out two Al Qaeda operatives, he said, but causes several civilian casualties too. The outrage caused by this collateral damage will inflame anti-US sentiment in Pakistan, and some of that sentiment will translate into support for Al Qaeda.  Some of that support will further translate into some people actually joining Al Qaeda.  Even if only 0.1% of Pakistanis are sufficiently outraged by the drone strike to support Al Qaeda, and only 0.1% of them will actually join the organization, that is still 170 new recruits (since Pakistan has a population of 170 million).  So the net gain for Al Qaeda is 168 terrorists (170 new recruits minus 2 casualties).

Of course, the precise figures will depend in part on the ratio of terrorists killed to civilian casualties, and there is some disagreement about this.  In his excellent book, Task Force Black, the BBC journalist Mark Urban claims that “most of the people killed in these attacks are innocent bystanders” (p.xiv, Foreword to Paperback Edition,2011). But The New America Foundation, a Washington think-tank, estimates that 80% of the 2,551 people killed in drone strikes since 2004 were militants, rising to an astonishing 95% in 2010. The expected utility of drone strikes could depend crucially on who is right.

Even if the drone strikes mainly kill militants, their utility may also depend on whether these are terrorist chiefs or low-level fighters. According to the New America Foundation, out of the 2,551 deaths, only 35 were recognized militant chiefs, or just 1.3% of the total. Among these high-level hits was the fearsome leader of the Pakistani Taliban, Baitullah Mehsud, who was the country’s number one public enemy. Nevertheless, the vast majority of targets have been low-level fighters.

There are also disagreements about local perceptions.  While those in Islamabad might react with anger, many locals privately support the strikes against extremists who have overrun their homeland.  What may inflame anti-US sentiment more than the civilian casualties in themselves, however, may be the lack of transparency and accountability. There is no investigation of civilian casualties, and no compensation paid, and some question the legal basis of the attacks.

It is probably too early to be sure whether the drone strikes are harming al-Qaeda and related groups, or spurring on Afghanistan’s powerful insurgency.  But my friend’s argument does at least set out the terms in which we should be thinking about this question, and suggest the kinds of data we need to collect before we can reach a definitive verdict.  How, for example, can we measure anti-US sentiment? And what objective metrics might we use to judge overall success?  One such metric might be the number of suicide attacks in Afghanistan.  Unfortunately, on this measure, the drone strikes do not appear to be achieving their aim; over the past year, the number of suicide attacks in Afghanistan, many launched from Pakistan, has soared.

Animal combat

When an up-and-coming chimp is deciding whether to challenge the alpha male to a fight, and when two antelopes size each other up before locking horns, each combatant must estimate his chances of winning.  If they overestimate their chances, they’ll regularly get a beating, and if they underestimate their chances, they’ll lose out on the rewards of success (which usually includes lots of sex).  So natural selection has probably honed a certain degree of risk intelligence in animals where males engage in violent forms of intraspecific competition.

Fighting antelopes

Two Gemsbok antelopes (“Oryx Gazella”) having a go at each other

I wonder if anyone has collected statistics about the outcomes of these fights.  If so, I’d be curious to know whether challengers win about half the fights they start.  If they win substantially more than half, or significantly fewer, this wouldn’t necessarily mean they were underestimating or overestimating their chances of winning, because we don’t know how the costs of losing and the benefits of winning stack up against each other.  If the benefits of winning outweigh the costs of losing, then a good utility maximizer would risk starting fights even when the odds were stacked against him.  The costs and benefits might be different for challengers and  incumbents, of course, which would make things even more complex.  Someone must have modeled this, and there’s probably some complex game-theoretic analysis of optimal behavior, but are there any empirical data?  If you know of any, please email me at