The Scales of Justice and the Balance of Probabilities

Win more and lose less by using decision and risk analysis

by Lefki Giannopoulou and Gavin Lawrence

Litigation risk analysis alone may maximise expected value but will not necessarily produce the best decision

For all businesses, litigation is a risk. It has significant upside and downside implications. Litigation is always time consuming, stressful and – for some – expensive. It has been amply demonstrated that businesses and lawyers who develop a comprehensive risk management approach to litigation can avoid costly drawn-out trials or arbitration proceedings – or, conversely, can profit accordingly.

Increasingly, legal practices are taking on-board basic risk management processes, driven by conditional fee agreements. Corporations, in combination with their legal representatives, are taking a decision analysis approach to prospective litigation, settlement, arbitration, pre-trial, trial and appeal.

Businesses need a quantitative analysis to assure themselves that any legal strategy adopted is likely to yield the highest expected monetary value (EMV). However, the best business decision is unlikely to be based on quantitative risk analysis alone.

Broadly, litigation can be expected to impact on creditors, lenders, customers, investors, stock market analysts and employees of both claimant and defendant. To bring together such divers issues requires the use of Decision Theory based analysis.

Decision making

Decisions are made to achieve goals. Decisions are based on beliefs to the extent that they involve uncertain options and unknown outcomes. Each alternative is associated with a choice of probability distributions. When, as in many legal actions, the probability distributions are unknown, a "decision under uncertainty" occurs. It will be appreciated that to develop a litigation-negotiation strategy complex maths will be required.

A company embarking on litigation should require their lawyers to answer fundamental questions:

  • What is the chance of winning?

  • What will the judge award in damages?

  • What will the costs be if the case is lost?

  • Should we negotiate and, if so, for how much?

  • How much should we spend to pursue/defend the case we are facing?

  • What resources should we allocate to each stage of the case?

  • What strategies will achieve our objectives?

In order to produce business-focussed answers to these questions, lawyers must adopt a risk and decision analysis process.

Risk Analysis and Decision Analysis both have their roots in Decision Theory - a body of knowledge and analytical techniques of varying formality designed to facilitate a choice among a set of alternatives.

Decision Analysis techniques based on Decision Theory can lead to better decisions by clearly delineating the problems faced. With a clear understanding of the structure of the problem and the uncertainties and trade-offs inherent in all the alternatives and possible outcomes, the lawyers and the problem owners can:

  • Negotiate more effectively

  • Control the resource allocation

The decision analysis process

A good decision is one that is made on the basis of a thorough understanding of the problem and a natural strategy would be to "divide and conquer", in order to simplify an apparently complex task. These smaller parts can then be brought together to create an overall representation of the decision situation, so that a course of action can be made apparent.

The approach that is followed in Decision Analysis is depicted in Figure 1. The first step is to identify the situation and to understand objectives within that situation. When the objectives are identified, the next step is to explore alternative courses of action and probable outcomes. Given the defined objectives then the outcomes can be evaluated and the uncertainties assessed.

Figure 1: Decision analysis process. 2

The next two steps consist of decomposing the problem structure in order to isolate and deal with the uncertainties and values associated with it. Sensitivity Analysis is done on the completed model. During this stage, the aim is to examine the implications of one or more aspects of the model and to determine whether they affect the projected optimal decision. In this interactive process, the decision maker’s perception of the problem changes. Beliefs or assumptions about various eventualities may develop, along with preferences for outcomes not previously considered.

Structuring the decision situation

The first step in solving a legal problem is to identify the individual elements, values and objectives, decisions, uncertain events, and possible consequences.

Values and objectives: The term ‘value’ is used to refer to things that matter to the decision maker. The term ‘objective’ means that which he/she wants to achieve. Litigation in business usually involves only one primary objective - money - but can involve others such as establishing a precedent or even revenge!

Decisions to make: With the context understood and values and objectives acknowledged, the lawyer and the decision maker can begin to identify specific elements of the decision to be made. Clemen 2 argues that it is important to recognise that many situations have as a central issue a decision that must be made right away and that one decision will usually lead to another in a sequence. Identifying the immediate decision that must be made is a crucial step. Future decisions will depend on what has happened before.

Uncertain events: Many important decisions have to be made without knowing what the precise outcomes will be. This uncertainty can be taken into consideration in sequencing the decisions still to be made.

Consequences: A consequence can be defined as all that follows as a result of pursuing an objective. Decision makers usually think about the consequences becoming evident at the end of the time line after all decisions have been made and all uncertain events resolved. This is the point at which people consider whether they have made a profit or a loss - the primary objectives in the court cases we are dealing with. The end of the process is profit or loss. The problem and its consequences can be measured in monetary terms.

Modeling the court outcome

After identifying the structure of a legal dispute process, a model must be developed to compare with the negotiated strategy outcomes and which will show the probability of a specific award when a dispute is taken to court or arbitration. For this purpose a flexible generic model is used.

In the generic model, the ‘judge’ may appoint an award to the claimant after hearing the evidence presented by both parties. In the model the claimant’s counsel will have presented evidence, which will, in turn, have been examined by the defendant’s barrister. The defendant’s barrister then presents counter-evidence, and so on.

The Decision Analysis approach to such a process is to help the decision maker and the lawyer predict what the judge will do. The whole task is divided into smaller parts (decomposition) and separate estimates of probability distributions are ascertained. The next step is to determine the combined effect of all these elements using Bayes’s Theorem and to predict a probability distribution that will apply to the judge’s decision.

Bayes’s Theorem is a mathematical formalisation of the process of learning from experience 6. Put simply the theorem says that one’s posterior belief is directly proportional to one’s initial belief when modified by new data:

Prior belief x likelihood = Posterior belief

Using Monte Carlo Simulation

Although the quality of evidence in a court model is subjective, it can be expressed as a probability distribution. The probability of a judge’s award can be examined with Monte Carlo Simulation.

Monte Carlo Simulation is an essential tool for developing a model of uncertainty. Winston 7 states "Monte Carlo simulation is the simulation where the random numbers used for each trial are analogous to a spin of the roulette wheel at a casino. Like the spins of a roulette wheel, the random numbers used to generate demands for each trial are independent." Monte Carlo simulation involves the use of a computer to generate a large number of possible combinations of circumstances that might occur if a particular course of action is chosen.

Graph 1. CDF of damages awarded by judge in a specific case

The cumulative density function output of a Monte Carlo simulation in the case depicted in Graph 1 shows the expected monetary value (EMV=average) of damages awarded to be 317,571. The range of awards in the simulation is between 0 and 1.2m with a 95% confidence limit of 1m or less.

Using decision trees and influence diagrams

After having calculated the probabilities of the awards made by the judge to a claimant for each specific case, the next step would be to evaluate the strategies to resolve the dispute. In order to accomplish this, we build a decision tree.

Decision trees can help decision makers develop a clear view of the structure of a problem 4 and, as a result, make it easier to determine the possible scenarios that can result if a particular course of action is chosen. Usually, decision problems are multi-stage in character, since a choice of a given option may result in circumstances that require another decision to be made.

Decision trees show all possible decision options and chance events with a branching structure. Decision trees proceed chronologically, left to right, showing events and decisions as they occur in time. In a tree diagram, the number on each branch of the tree is the probability of following that branch, given that one has reached the root of the branch in question (the roots are on the left). To calculate the probability of getting to the end of a branch, on the right, all probabilities along the way must be multiplied. Then by working backwards from the final nodes, we assign an expected monetary value to each node and show at each choice node which decision is best. An example of a negotiation decision tree is shown in Figure 2 below.

This kind of diagram helps people think of all the possibilities 1. It is especially useful when the probabilities of the various branches are influenced by different sorts of factors, so that some of these probabilities might be constant over a variety of circumstances that affect the others.

There are drawbacks to the use of decision trees in so far as the number of alternatives can grow almost exponentially, making them difficult to follow or to validate. Continuous pruning is required to keep them under control. An alternative approach is to use Influence Diagrams, which are isomorphic with decision trees but are easier to understand and show the relationship of all the elements of the problem. Furthermore they can cope with substantial sophistication and complexity. Figure 3 is an example of a litigation model.

Again Monte Carlo simulation of the negotiation process is used to determine the value of the outcome. This can be compared with the output of the trial simulation in order to decide at what point to end negotiations and push for trial.

Making the best decision

The best decision is the one that most meets the needs of the decision maker in all possible ways, i.e., it is an optimal fit solution and not necessarily the optimised (EMV based) solution found in the quantified risk analysis (QRA). Graph 3 shows three generic profiles: the risk seeker, the risk avoider, and the risk neutral decision maker.

Graph 3. Generic Risk Profile

The risk profile determines the decision makers’ tolerance of uncertainty to achieve a desired objective.

The primary approach for determining the best decision is that of called Multi-attribute Utility Theory. In this technique all the attributes of the possible outcome are ranked and weighted in a cost-benefit analysis to determine the best fit decision. The process will usually involve a facilitated decision workshop for which a wide variety of software packages are available.

For a person who is risk neutral, maximising expected monetary value (EMV) is the same as maximising Utility. For large corporations that make decisions involving amounts which are small relative to their assets, it is reasonable to assume a risk neutral profile. The Utility gained from winning or lost from not winning is a function of the litigants’ wealth; therefore, for a business, it is a complex function that includes the firm’s assets, profitability, future cash flows, time horizon and business context as well as the personal criteria of the decision maker and other non-monetary issues. When modeling this, the EMV is first determined and then its sensitivity to various risk profiles.

Utility Theory remains almost unchanged – complete with its faults - from its original conception by von Neuman and Morgenstern in 1947. Substantial empirical research shows that decision makers, under certain circumstances, make illogical choices. Other theories have been developed to explain the illogical decisions, e.g. Prospect Theory, but given the limits of Utility Theory it remains the most practical and useful tool for determining the ‘best’ decision at present.

An alternative approach that has found favour with both decision analysts and behavioural economists is that of Regret Theory5. Regret is defined as the psychological reaction to making the wrong decision where wrong is determined by actual outcomes rather than in relation to the information available at the time the decision was made 3. As with utility maximisation, this methodology incorporates the risk profile of the decision maker. To-date, practical applications of Regret Theory and disappointment models appear limited, although some success has been reported in the financial sector with portfolio selection. It should be expected that this would change in the near future as developments in regret modeling advance significantly.

The best decision, whether based on either the Utility or Regret theories will, in all likelihood, be significantly different from the decision based on EMV alone, because with the use of risk and decision analysis it is possible to incorporate and evaluate far more issues than traditional quantitative risk management approaches permit.


Identifying and structuring the uncertainties of a case decision situation according to the above analysis can provide lawyers and decision makers with a better understanding of the issues they face and show to them the best decision to reach their objectives -- money or otherwise. The benefits of this approach can be summarised as:

The Benefits of Litigation Strategies and Risk Management

  1. Identification of key factors that influence outcomes
  2. Identification of biases and explicit disagreements and evaluation of their importance.
  3. Quantification of the risk and uncertainty inherent in the key factors.
  4. Cost-benefit relationship of various scenarios and activities.
  5. Determination of likely settlement value at the outset of litigation.
  6. Calculation of the probability of success and failure for each potential strategy.
  7. Assessment of the likely costs of each stage of the action.
  8. Inclusion of insurance coverage issues with all the other decision-making issues.
  9. Multi-attribute utility or regret theory based evaluation of complex financial and non-monetary criteria.
  10. Determine the negotiation - trial saddle-point.
  11. Presentation of complex legal and factual issues in a simple graphic format.
  12. Unambiguous communication between clients, management, lawyers and experts.
  13. Simplification of complex case structures.
  14. Single comprehensive picture that captures all issues.
  15. Clear understanding of the impact of early settlements and trial outcomes on overall strategy decisions and the final outcomes.
  16. Effectively influence settlement using the information provided by these decision and risk analyses.
  17. Justification and validation of all decisions in clear explicit terms.
  18. Hard copy confidence in the negotiation and legal strategies.

This analysis is neither intended to replace the decision-maker’s or the lawyer’s intuition, nor to relieve them of obligations in facing a litigation; or to compete with their personal style of analysis, but to complement and augment.


  1. Baron, J. (1994). Thinking and Deciding (2nd edition). Cambridge University Press.
  2. Clemen, R. (1996). Making Hard Decisions: An Introduction to Decision Analysis (2nd Edition). Duxbury Press.
  3. Dembo, R. S. and Freeman, A. (1998) Seeing Tomorrow. John Wiley
  4. Goodwin, P. and Wright, G. (1997). Decision Analysis for Management Judgment, (2nd edition). John Wiley.
  5. Loomes, G. and Sugden, R. (December 1982) Regret theory: an alternative theory of rational choice under uncertainty. The Economic Journal, 92 pp805-824.
  6. von Winterfeldt, D. and Edwards, W., Decision Analysis and Behavioural Research, Cambridge University Press, 1986.
  7. Winston, W.L. (1996). Simulation Modeling Using @Risk. Duxbury Press.

Lefki Giannopoulou BA., MSc.

Born in Athens, she lived for three years in Brussels and traveled extensively around Europe. She studied Economics at the University of Athens where she received a First Class Honors Degree.

Challenged by the obscure nature of Economics, she decided to try to explore fields, which could provide her with the means and the ways to tackle this uncertainty. Decision Sciences appeared to be the most appropriate subject of study, thus the postgraduate degree in Decision Sciences from the London School of Economics (2001).

Now, she is planning to examine how modeling and simulation, as instruments of dynamic analysis and risk assessment, combined with other inputs, can lead to right decisions to different kind of problems, i.e. law, business, finance.

Gavin Lawrence FRSS., BA., MSc., MBA.

Having started his career in advertising he developed a fascination for the purchase decision process of both consumers and organizations - particularly on how to leverage the purchase decision process.

The primary focus of his work has been specifically on decision making under uncertainty and the maximization of utility. His subsequent work has addressed joint decision making, gender differences in decision making, reconciliation and negotiation strategies to optimize the joint decision outcomes.

He has worked for a variety of companies and consultancies in the UK and USA working in many areas of risk and decision analysis ranging across project, business and financial risk in sectors as diverse as railway infrastructure, oil wells, venture capital funding and PPIP/PFI projects.

Gavin P. Lawrence provided permission to publish and store this article at DSSResources.COM on Thursday, May 23, 2002. This article was posted at DSSResources.COM on July 19, 2002.


Giannopoulou, L., and G. Lawrence, "The Scales of Justice and the Balance of Probabilities", DSSResources.COM, 07/19/2002.