# We Need To Talk About Ergodicity

You have a gun which holds six bullets, but only has one in the chamber.  You use it to play a game of Russian roulette with a group of 19 other people.  Each of you takes one turn in spinning the chamber, holding the gun to your temple and pulling the trigger.  If you are successful you win £1m, if not, well, then you die. Whilst this may not be an appealing proposition, your chance of death is relatively low (17%), and potential for becoming a millionaire high (83%).  It is also far more attractive than an alternative version of the game where instead of playing with a group, you play on your own.  In this instance there are still 20 turns but each time the gun is directed at your head.  The odds on the outcome for you in this instance are not so favourable.

These contrasting approaches to Russian roulette are a typical example of ergodicity[i] [ii]. A system is deemed ergodic if the expected value of an activity performed by a group is the same as for an individual carrying out the same action over time.  Rolling a dice is an example of an ergodic system.  If 500 people roll a fair six-sided dice once, the expected value is the same as if I alone roll a fair six-sided dice 500 times.

The Russian roulette example is a non-ergodic system.  The expected value of the group differs sharply to the average of an individual carrying out the action through time.  In the group situation the average outcome is to live and become wealthy.  As an individual performing the activity through time – on average – I am dead.  In a non-ergodic system the group expected value is deeply misleading as it pertains to individual experience.

Although these may seem like somewhat frivolous examples, the concept of ergodicity is incredibly important.  Much of classical economics assumes about human behaviour is founded on the expected average outcome of the group (see Expected Utility Theory).  This works under the assumption that most environments or situations are ergodic, when in fact this is not the case.  The best starting point for understanding ergodicity economics is this article in Nature, by Ole Peters[iii].

Given the implications for classical economics, the idea of ergodicity is also incredibly important for behavioural economics.  Many of the ‘biases’ identified in this field are expressed as violations of the assumptions made in classical economics and therefore deemed irrational.  Yet what if the starting assumptions are incorrect in the first place? What if much of what classical economics says about decision making is based on the average outcome of a group; when my ‘rationality’ is best judged by considering my individual experience through time?

Ergodicity is not the most intuitive concept, so let’s take another example – home insurance. If we assume that insurance companies make a profit from selling us insurance on our houses, then surely it doesn’t make sense for anyone to buy insurance for their home?  The insurance company is making a profit, those buying insurance must be making a loss.  Yet this negative expected loss applies to the group average, the situation is different for the individual.  The experience of the group is irrelevant to me as an individual.  What I care about is the impact on my wealth through time – there is only one of me.  The risk of ruin from my house burning down is what matters.

The concept of ergodicity is also critical when thinking about major issues such as inequality. Let’s take the standard economic measure of economic growth – GDP – what does that tell us about individual experience? Very little[iv]. GDP is a group measure.  Therefore we have another situation where the average outcome of the group can be very different to any individual’s experience.  We could therefore enter a situation where economic growth numbers (measured by GDP) appear impressive, but they mask the fact that inequality is burgeoning – wealth is accumulating to a select, small group, whilst more individuals suffer (surely this couldn’t actually happen!?).  Focusing on extreme cases of success in non-ergodic systems can be incredibly deceptive.

Ergodicity and Behavioural Economics

Given the subject matter of this blog, it is perhaps worthwhile exploring a couple of examples of where the concept of ergodicity has implications for ideas in behavioural economics.  One of these was explored by Jason Collins.  In his blog[v] he looked at the following scenario, which draws on work from Ole Peters and colleagues:

“Suppose you have \$100 and are offered a gamble involving a series of coin flips. For each flip, heads will increase your wealth by 50%. Tails will decrease it by 40%. Flip 100 times.”

This type of bet is often rejected by individuals, despite the expected gain being 5% of wealth at each flip.  Declining this type of bet is often put down to risk aversion.  But is turning down a bet with a positive expected value such a bad idea?

Collins ran a simulation of 10,000 individuals flipping the coin 100 times each. Whilst the average wealth reached \$16,000, the median was only 51 cents. 86% of the population saw their wealth decline.

A bet that looked good on average actually led to catastrophic outcomes for most, whilst a select, fortunate few made huge amounts of money.  Again, the average outcome of the group was meaningless to most people.  And, as Collins goes on to explain, if you increase the number of coin tosses eventually everyone will end up financially ruined.

There is inevitably a lesson here for investors about the destructive power of negative compounding. And from a behavioural economics perspective there is also a valuable insight into why seemingly irrational decisions (turning down a bet with a positive value on average) can be viewed as rational when considering the experience of a given individual over time.

Probability Weighting

Ergodicity can also matter when we consider how we ‘weight’ probabilities when making decisions.  In a recent paper Ole Peters and colleagues[vi] explored a key tenet of cumulative prospect theory – that people overweight the probability of rare events with extreme outcomes.

The authors argue that far from being an error of judgement, a propensity to ‘exaggerate’ the likelihood of low probability extreme events is a reflection of greater uncertainty.  Individuals have less information about uncommon events (their historic frequency is low) and therefore a greater potential for error in their assumptions.  This feature allied to the risk of ruin from an extreme event means that adopting a cautious approach to such likelihoods is prudent.

The idea of increased uncertainty about probabilities for rare, ruinous events is a compelling argument as to why a seeming overstatement of probabilities may not be an error of judgement.  There are, however, other pertinent issues to consider around how individuals gauge probabilities.  Rather than simply overstate the likelihood of extreme events, there is evidence that in certain circumstances individuals ignore certain high impact risks, seemingly applying a zero probability weighting to them.  Kunreuther’s work on insurance showed that individuals often don’t buy disaster insurance until after they have experienced a loss from such an event[vii].  My contention would be that an individual’s judgement about the probability of a low likelihood, high impact event is intertwined with its salience and availability.  We are less likely to ‘overstate’ the probability of an event if we have never observed it, or if it lacks any emotive qualities.  Our perception of risk needs to be over some ‘threshold’ for us to consider it at all.  We can’t worry about everything.

What Are the Implications for Investment Decisions?

There are a range of areas where considering ergodicity could influence investment making; not least in the sizing of ‘bets’ and the potential use of the Kelly Criterion, but that will be for future posts.  A simple example of where ergodicity might be important is in portfolio construction.  Let’s assume we are allocating to ‘alternatives’ in our portfolio and have modelled them on the basis of them delivering a 4% annualized return with 7% volatility (remember this is hypothetical).  Is the average expected return of a group of hedge funds that meaningful?  Not really.  This is a non-ergodic system – we are interested in the path of returns for the individual funds that we select; the average result of the group might match our forecasts (unlikely), but we could still end up with very poor results.

Concluding Thoughts

Given that the field of behavioural economics was forged on the identification of limitations in the assumptions of classical economics; it seems reasonable that ideas central to the amorphous field of ‘behavioural economics / science / finance’ should also be held up to scrutiny.  Whilst we should always be cautious about attempting to identify ‘one big idea’ that explains everything; the concept of ergodicity is crucial lens through which we should be observing decision making.  We need to talk about it more.