Game theory and the art of litigation settlement
This article is the first in a series targeted at litigators that consider the issue of settlement in litigation through a game theoretical lens.
Article 1 – graphically mapping the settlement acceptance zone
Opening remarks
Game theory is a branch of mathematics dealing with conflict situations where a participant's choice of action depends critically on the actions of other participants. This series aims to present the key concepts and insights that emerge from this analysis in a graphical format, without complex mathematical formulae. It is hoped that this approach will provide litigators with a new perspective and framework within which to consider settlement.
There is one point to address at the outset. It can be fairly pointed out that the type of analysis advocated in these articles is predicated on inputting assumptions about one's own case and one's opponent's mind-set. That is a fair criticism and it is accepted that the value of this form of analysis is only as good the judgments of the litigator conducting a case and even then there is considerable uncertainty. Notwithstanding that caveat, the insights and observations that emerge from deploying a probability based assessment to settlement are valuable in terms of testing the timing and quantum of offers which are routinely made in litigation.
Summing over probabilities
First, a quick crash course on probability.
Suppose you have a coin which you propose to flip and you say "heads I pay you £10 and tails I pay you nothing". How much is that opportunity worth? The answer seems obvious; the opportunity is worth £5 because you have a 50% chance to be paid £10. In arriving at that conclusion you will have, in mathematical language, 'summed over the probabilities' – i.e. you will have added together the 50% chance of winning £10 for a heads flip to the 50% chance of winning £0 on a tails flip.
Taking another example, suppose you have a six sided die and you offer to pay someone in pounds whatever number you roll with it. How much is a roll of that die worth? The method is the same as with the coin toss – you multiple the probability of each outcome with the value of that outcome and add them together, i.e.:
What is the value of a legal case?
With that technique we are now in a position to calculate how much a hypothetical legal case is worth. At a basic level, assuming a binary outcome, suppose a claimant is suing a defendant for £1 million and the claimant thinks it has a 50% chance of winning £1 million and a 50% chance of losing; in that circumstance the claimant could value its claim at £500k (i.e. 50% x £1m + 50% x £0m).
This value is of course simplistic as in reality the value of a legal claim is complicated by the fact that one has to pay lawyers (and incur various other litigation costs) and some of those costs would potentially be recoverable from the defendant (were the claimant to win) or the claimant might have to pay a proportion of the defendant's costs (were it to lose).
The claimant indifference line
Let us park the issue of cost recovery for now and focus on the impact of spending money on legal fees. Consider for a moment a case that proceeds to the end of trial, during which time the claimant incurs various legal costs. At what value would the claimant theoretically be willing to settle prior to hearing the result of the trial? The answer is £500k. That is the claimant's expectation of the average value of what the court would award (i.e. 50% x £1million + 50% x £0m). Perhaps counterintuitively, the amount of legal costs the claimant has spent is irrelevant to this answer. That is because they are – as an economist would say – a sunk cost.
However, although legal costs are irrelevant to the value at which the hypothetical claimant would settle the case (using a probability analysis), they are very relevant to the claimant's overall net recovery. Let us assume that to litigate the case to the end of trial would cost £300k. In that case the claimant's overall net recovery, if it settled on the last day of trial would be £200k (i.e. the £500k settlement minus £300k in legal costs).
Let us assume that the hypothetical defendant in this case made a settlement offer to the claimant at the outset (i.e. before any meaningful costs were incurred by the claimant) of £200k. Should the claimant accept the offer? For simplicity in answering this question we will ignore the impact of interest rates and inflation on our decision. The answer is (thinking purely in monetary terms) that the claimant should not care either way as the claimant ends up with an overall net recovery of £200k regardless.
We can show this by plotting these two scenarios as two points on a graph showing settlement value against the claimant's incurred costs.
Figure 1 – Plot of the claimant indifference line for a case where the claimant believes it has a 50% chance of being successful at trial (assuming a claim value of £1 million and expected legal costs of approximately £300k to trial)
Crucially, we can observe that the claimant's indifference extends to any point along the line connecting these two points because anywhere along this line it would obtain the same net recovery. To illustrate, consider the following points along the line:
Settlement value Legal costs Claimant net recovery
£0.2m £0m £0.2m
£0.3m £0.1m £0.2m
£0.4m £0.2m £0.2m
£0.5m £0.3m £0.2m
Self-evidently the claimant would like to settle for the highest amount it can. This desire translates graphically as wanting to settle as high above its indifference line as possible. The claimant would not want to settle below the indifference line as that would be less valuable to it than its expected recovery at trial.
The defendant indifference line
Let us now consider our hypothetical case from the perspective of the defendant. For simplicity, we will assume that the defendant shares the claimant's view of the merits of the case and would incur broadly the same costs defending the case through to trial. As the defendant's view of the merits of the case matches that of the claimant, the defendant would, like the claimant, be willing to settle the case for £500k i.e. the amount of its expected liability. As with the claimant, the defendant would be indifferent to settling the case at this point or earlier as long as its net loss outcome was the same. We can show this graphically in the same way as we did for the claimant.
Figure 2 - Plot of the defendant indifference line for a case where the defendant believes it has a 50% chance of being successful at trial (assuming a claim value of £1 million and expected legal costs of approximately £300k to trial)
The downward trajectory of the defendant indifference line reflects the fact that the defendant would be willing to pay a higher settlement value at the outset to avoid incurring legal fees as opposed to the claimant who would be willing to accept a lower settlement value to avoid the same legal fees.
The settlement acceptance zone
The graph below plots the claimant and defendant indifference lines together. We have previously established that the claimant would wish to settle as high as possible above its indifference line and for the same reasons the defendant would wish to settle as far below its settlement line as is possible. The shaded area shows the space, which we shall refer to as the 'settlement acceptance zone', above and below the claimant and defendant indifference lines respectively. This area maps out the theoretical space where the claimant and defendant could both agree a settlement more or equally favourable in both their eyes to litigating to trial.
Figure 3 - Settlement graph where both parties believe they have a 50% chance of being successful at trial (assuming a claim value £1 million and both parties have expected legal costs of approximately £300k to trial)
What we can immediately see from the graph is that the range of theoretically possible settlements is widest at the outset of the case and shrinks as costs are incurred as the matter progresses to trial.
In our example the claimant could theoretically settle for £800k (the highest point in the settlement acceptance zone), which would be £300k more than its expectation of recovery at trial and which would save it an additional £300k in legal fees as well (i.e. a net improvement of £600k). Conversely the defendant could theoretically settle the matter at £200k and make the same level of saving. In reality, assuming equal bargaining positions, one would expect the claimant and defendant to settle at the mid-point between these positions (i.e. £500k). If they were to do so they would both be better off by £300k than their respective expectations of recovery/liability at trial.
As noted, we have disregarded from this initial analysis costs that either the claimant or defendant could potentially recover from the other if successful. In this particular example, by design, including them would not impact the result as the two cancel each other out (on average) given the symmetry in the parties' legal spends and anticipated prospects of success.
Conceptually, in this example where the claimant and defendant broadly agree on the merits of the case, the settlement position reduces to the claimant and defendant negotiating how to divide the combined legal fees that they would otherwise have spent to take the matter to trial. Of course, as a case progresses and the assessment of the merits evolves, then the lines may shift relative to each other, giving rise to an evolving settlement dynamic.
Summary
In this article we have considered how to map the settlement prospects of a straightforward legal case graphically and observed for our example case that there is considerably more scope to settle the case earlier rather than later. We have also seen in this particular example that the settlement position can be thought of as a negotiation of how to divide the pool of legal costs that would otherwise be spent by the parties taking the matter to trial.
In the next article we will consider how the position changes where the claimant and defendant do not agree on the merits of the case.