Calculating Expected Value, but Beware the Ludic Fallacy

by Bruce Greig

Parties rarely come to mediation with a clear understanding of what their best and worst outcomes at court really look like.

For a party claiming £250k, they initially come into the claim thinking “Best case, I get £250k, worst case I get nothing.”

After an initial conversation with their legal team, they sort-of pay attention to something you said about costs, and think “Best case I get £250k, worst case I lose out a bit on legal fees”.

It is common for parties to not go much further than this in calculating their best case and worst case. But I would say that there are three further steps to take in the process (and one step back at the end):

First, bottom out the initial calculation, by fully costing in legal fees both ways

Everyone probably knows how to do this step, but it is surprising how often parties come to mediation without having done it.

In a £250k claim there could be £50k of costs to go to trial, on each side. If the claimant wins, they get their £250k, less their £50k costs, plus perhaps 70% of those costs back on standard basis costs award. So their best case is £250k - £50k + £35k = £235k.

Their worst case is £50k of their own costs, plus £35k contribution to the other side’s costs, so their worst case is a loss of £85k.

Their range of outcomes was £0 to +£250k, and is now -£85k to +£235k.

The second step is to calculate expected value

Even fewer parties take the trouble to calculate expected value. Expected value is a simple maths concept where you multiply the value of each outcome by its probability. The expected value of a raffle ticket with a 5% chance of winning a £1,000 prize is 5% x £1,000 = £50.

If you estimate that your client has a 70% chance of being up £235k and a 30% chance of being down £85k, their overall expected value is:

70% x £235k = £165k

30% x £(85)k = £(26)k

Net: £165k - £26k = £139k

So their £250k claim has a net expected value of £139k.

Another way to look at this is if you could run this trial one hundred times, you’d win seventy times, netting a total of £16.5m. You would lose 30 times, paying out a total of £2.6m. Your net winnings would be £13.9m, or £139k per case.

The final step is to run through all this for the other side, from their point of view

You might not know exactly what their costs are, or what they think their chances of winning are, but you can make a reasonable estimate.

Armed with these figures, you can see what the parameters for a negotiated settlement might be.

But, beware the Ludic Fallacy

A note of caution. Beware of what Nassim Nicholas Taleb calls the ‘Ludic Fallacy’: the over-reliance on probabilistic thinking in real-life situations.

In a casino you know exactly what the odds of landing on red are. In a courtroom you do not know this. These calculations are useful to set up some parameters to help guide a negotiation. But, unlike in a casino, your client does not get to roll the dice many times and average out their wins and losses.

Some people argue that expected value is completely useless for making an actual decision in real life, because an individual outcome is almost certainly going to be much larger, or much smaller, than the expected value.

Even if the expected value is +£5m, there is no actual outcome which results in your client taking home £5m. The possible outcomes are all either much bigger than £5m, or much lower than £5m. Philosophically, what does it even mean to say that a single outcome from a single event has a probability of 70%? It either happens, or it does not happen. It cannot happen 70% of the time, because it will only happen once, or not at all. Caveat lusor, player beware.

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