The Sunk Cost Dilemma

1-player game theory helps understanding why projects run into troubles

Some background on game theory

The findings of Game Theory have often been found helpful for the description of business situations in which individuals or groups have interests which are contrary to those of a group to which they belong. The following example describes a situation where a dominant strategy leads to a dilemma without such a conflict of interests.

Games are defined as follows:

A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies.
(see http://en.wikipedia.org/wiki/Game_theory)

Depending on the number of players, a game may be categorized in one of three types:

2- and n-player games may be simultaneous (one match at a time) or sequential (the game is made by consecutive moves). They are by nature competitive and the motivation of a player is that to win.

1-player games are always sequential. These games are characterized by the player's desire to meet a set of objectives. The motivation to play is achievement³ or reward for achievement.

Very much simplified, one may say that

Game Theory for 2-player games has been described by John von Neumann and Oskar Morgenstern (1944, Theory of Games and Economic Behavior, Princeton University Press).

n-player games have been described by John Nash in his dissertation Non-cooperative games (1950) with the invention of the Nash Equilibrium. A similar approach could be found in the article The Tragedy of the Commons by Garrett Hardin (1968).

To the author's understanding is this document, which has been originally published on 12 May 2006, the first approach to describe game theory for 1-player games.

The approach

Another situation will be described here: It is a common dilemma situation which applies to project management and may be described as the Sunk Cost Dilemma.

It can be seen as a one-player game in which a single player (which may be a group which in our context is not characterized by internal competition) makes a number of consecutive moves like in the card game Solitaire (or Patience) or in the game called Rubik’s cube which was popular around the world in the early 1980ies.

The Sunk Cost Dilemma can apply to any game with 1 player and n moves (n>1). The player calculates and optimizes before each move which option out of a set of two or more options is the best and implements this option, but finds at the end that the sum of a sequence of optimal decisions may be that of one big bad decision.

The Sunk Cost Dilemma is different from Repeated Games in that it applies a 1-player approach whereas Repeated Games are played by two players.

The following case study will show that linear elements (running costs of a project) can add to a Sunk Cost Dilemma situation as well as non-linear elements (incentive payment, liquidated damages).

Definitions

Sunk costs

The term sunk costs is being used here as a descriptor for cost amounts which are no more under control, i.e. they are no more open to decision making. They can not be rejected, reduced or recovered because they have

• already and definitively been paid to other parties, or

• have been validly invoiced or charged and the invoices or charges will definitively have to be paid in near future, or

• have not yet been invoiced or charged but the invoices or charges are expected to come in in the foreseeable future, the amounts are already known and the payments will have to be settled soon after the invoices or charges have been received.

Sunk costs may differ from Actual costs which at least relate to payments already made. Actual costs may also include valid invoices which are already due to be paid, but should not include expected invoices or payments. Actual costs relate to actual status of payments while Sunk costs relates to decision making. If a payment can be validly recollected e.g. by applying legal measures, than these may still be actual costs before the re-imbursement has been actually made, but would not been perceived as Sunk costs.

Sunk costs may include

• indirect costs – costs for overheads, costs to maintain the production and the performing organization;

• direct costs – costs for input materials for production processes which are directly related to the scope of work.

Sunk costs will be typically calculated using monetary metric, but may also be shown in resource working time, availability time of human/equipment type resources, or in materials consumed. Other metrics like scrap produced or environmental damage occurred could also be thought of.

Dilemma

A dilemma is a situation with an uncontrollable tendency towards some kind of disaster. It may be unforeseeable or even foreseeable, in the latter case the term tragedy is sometimes applied too.

Uncontrollable does not necessarily mean that the tools are not present to avoid the disaster. It may not be in the single and most immediate interest of each player or of most of the players to avoid the disaster.

In many cases, foreseeability therefore does not change the general characteristic of uncontrollability of a dilemma.

A case study

The upfront situation

The company Global Project Solutions Inc. (GPS) is in negotiations to run a project for another firm called HAL Inc. which includes building and implementing a piece of industrial production equipment which will be made of different types of hardware and software. In order to secure the buyer from cost risks, the contract offered was a fixed price contract with a price tag at $1,000,000.

The equipment will be mostly made from standard components, but there will be some customization necessary which will be agreed upon as soon as both the customer and the contractor have a sufficient understanding about the details of the customization.

The customer also proposed to pay an incentive fee in case that the contractor would be able to meet a more aggressive delivery date. Global Project Solutions Inc. estimated that a time of nine months would be sufficient to finish and hand over the equipment, but they would receive an additional payment of $150,000 in case they succeed in doing the handover before the end of the 7^th month.

According to a requirement brought up by the customer, a term for liquidated damages⁴ of $200,000 has also become part of the proposal in case that Global Project Solution would cancel the contract before all work has been finished. The same payment would apply if the customer can prove that the contractor is unable to finish the project from technical, organizational or funding reasons.

The project should start immediately after the contract has been signed off by both parties. The customer has already signed the contract, now the contractor has to confirm the order with the signature under the contract.

Decision 1:

The Pre-sales department at Global Project Solutions Inc. has made some rough estimates and calculations. They assume that

• the nine month estimate on duration is perfectly reasonable,

• with the additional amount available from the incentive payment, they should be able to speed up the project and achieve handover in less than seven months and therefore in time for the incentive,

• the company’s command of their own technology is good enough to feel sure that the project can be finished and that the company will not want to step back from the contract before the customer’s requirements will be Fully met.

The Pre-sales department calculated own costs to build the equipment at $800,000. A fast-track project plan to schedule handover during the 7^th month would be feasible at $900,000.

The first decision would include three options:

*: Would include the incentive payment to the contractor

As there are no opportunity costs – other projects which can not be started because of this project and which would gain a higher profit – the best decision is to begin the project and go for the fast-tracked option. This decision has the highest yield.

GPS signs the contract with HAL Inc. and immediately begins with work on the project.

Decision 2:

Some days before the second month has been finished—a project manager had been assigned and a core team been built during that time, also some first work had been done—the first valid project schedule could be finished and was ready for acceptance by the management of GPS and by the customer too.

During that time some underestimated difficulties related to customization came up: Some standard components normally used by GPS have been found incompatible with interfaces at the customer’s site and need to be replaced with custom made parts. 

The sunk costs – invoices already paid, open invoices and expected invoices plus internal charges from other departments – are already adding up at $100,000, after 40,000 at the end of the first month.

The new handmade parts will increase the overall total cost estimate by 10%. An additional payment of $10,000 will be paid to subcontractors as an incentive if they can speed up their production and deliver the components before the end of the 4^th month. This adds up to a cost increase of $100,000.

Urgent meetings were held at the end of the second project month in order to decide whether the company GPS should go on with the project or inform the customer that they would withdraw and pay the liquidated damages.

During these meetings the discussion concentrated on a decision value: How much of the costs are still open for decision making and what is the payment from the customer related to these decisions?

The decision at the end of month 2 looks like this (grey background signals options):

*: Would include additional costs and incentive payments to subcontractors.
**: For liquidated damages.

The decision to proceed is more attractive than the decision to cancel the project, its yield is $450,000 better and so is its profit. The best decision is to go on with the project, and this decision is being made.

Decision 3:

End of the 3^rd month, the project seemed quite on track. Progress was fine and another $80,000 have been sunk. As everything goes fine, no new estimate for the rest of the project has been made.

End of the 4^th month, a major audit was being planned. The additional components have been delivered and the progress overall seemed fine too. Costs at a total value of $390,000 have been produced and the cost estimates of the project seem still reasonable. The decision to proceed was easy looking at the numbers:

Decision 4:

End of the fifth month came with a bad surprise: The custom made components fit perfectly to the interfaces at the customer’s site, bit were difficult to integrate with the plant made by GPS. Some further adaptations have become necessary which will cost another $20,000 and will take over a month. This would make it impossible to adhere to the 7^th month for handover and to get the incentive payment. An additional payment of $80,000 could speed up the adaptations to have them finished during the 6^th month.

Regular project progress consumed another $200,000 and besides of the problem with the handmade components everything is well on track.

There are again three options for decision 4:

*: Would include another $20,000 for adaptations
**: Would include another $100,000 for adaptations
+: Would no more include incentive payment

The second option looks best. A decision to further invest $510,000 results in $1,150,000 income, which computes to a payoff $640,000.

And: There is still a—very small—profit. The other options would yield lower payoffs and the project manager would have to explain the management why the project made an overall loss.

Still a chance to take.

Decision 5:

During the sixth month it became clear that the additional payments will not be sufficient to get the additional components reworked in time. The project can no more be finished by the end of the seventh month and the incentive will not be granted by the customer.

The project manager got under pressure from the management which she passed over to the team The project got further derailed regarding time and costs. Another $200,000 have been sunk, but progress slowed down and open costs had to be re-estimated $250,000 higher than expected the month before.

At the end of the sixth month, there were two options for decision 5:

Instead of making a profit of $250,000, the project made a loss of $500,000.

The sum of five decisions, each of them perfectly yield-optimized for a project business owner’s profit, has been a bad decision.

At the beginning, the contractor had the option to not run the project—at least not using the aggressive approach—and later on had the opportunity to terminate the project at least four times. But the longer the project took, the harder it got for the contractor to opt out. The longer a project goes, the more likely it is that it will be finished, even when it runs into a loss.

The decision yield—how much profit can I make from now on and how much do I have to invest right now?—did increase over most of the lifecycle. Each single decision had a positive yield, but while this yield got even higher, the sum of all these decisions has become a loss.

Analysis

Nash equilibrium and Sunk-cost equilibrium

Is there a set of dominant strategies in a Sunk Cost Dilemma situation, similar to those constituting the Nash Equilibrium in an n-player game?

As the single decisions are made to maximize single decision yields, the dominant strategy for each single decision in the example has been "yes - go on". And similar to the Nash equilibrium, optimized single decisions did not lead to a positive total.

The Nash equilibrium occurs in a mixed cooperative/competitive situation. The Sunk-cost equilibrium is the outcome of a learning process in combination with increasing sunk costs—costs that are no more under control, but add to total costs. The growing knowledge comes together with reduced influence.

From the third decision on, terminating the project would even have created a higher loss than the final loss of the entire project. But one would know this only from hind side...

Nash equilibrium contradicted Adam Smith's theory of the "invisible hand". Sunk-cost equilibrium contradicts the common sense expectation that if every decision is a good decision by itself, the outcome must be that people are doing the right thing.

Warning Signs

Are there any early warning signs that a project—or maybe even an entire project portfolio—may be heading to a total loss even when individual decisions are right?

One should be careful if the following has been found:

• High risks linked to a date or another non-linear element (like the incentive fee in the example) are a common element of the Sunk Cost Dilemma. “Almost meeting” such a target can mean that additional costs have been incurred but there will be no pay-off for that.
  

• Three-point estimation may have shown that the risks are high and that the likeliness of not making a profit is high. In the example, a one-point estimation approach has been used, which hides the risks behind the assumptions.
  

• Early identification of constraints may have been helpful too. In the case study, it may have been possible to take care earlier what happens if the new system does not fit to the customer’s legacies. 
  

• Trend analysis may sometimes be helpful too as it shows a tendency of a learning curve following behind the money burning curve instead of running ahead of it—what the Sunk Cost Dilemma actually is.
  

•  Low-margin and fast paced projects are far more at risk than high-margin projects. The risk of running into the Sunk Cost Dilemma grows with decreasing margins and reduced execution time.
  

• Existence of incomplete contracts.
  
  Incomplete contracts⁵ seem to be generally unavoidable, according to recent research of Contract Theory.
  
  An incomplete contract increases the risk of running into the Sunk Cost Dilemma because costs for the contractor may become visible only late, when the contract is getting more and more complete, i.e. showing customer's assumptions and expectations.
  
  Contracts and related documents like CSOWs⁶, CWBSs⁷, Scope statements etc. should be meticulously investigated for the risk of a Sunk Cost Dilemma.

Vicious circles

The case study shows how the Sunk Cost Dilemma may come with systemic vicious circles which can not be planned for but may be identifiable when they occur:

• The project is behind schedule and over budget.

• The project manager gets pressurized by management.

• The project manager routes the pressure onto the team.

• The team is loosing confidence and communications are suffering.

• Interfaces cause extended troubles.

• Progress slows down and costs are soaring.

• The project gets further behind schedule and over budget.

Solutions

The Sunk Cost Dilemma is often a result of wrong presentation and interpretation of correct data.

The following example exists in reality. It shows a display solution used on shop floor level in a manufacturing company and has been published in the company's customer magazine:

Can Decision Trees help?

Yes they can.

Insight Tree supports three-point estimations and shows at the right end of each branch the payoff:

Option2 is more attractive with an EMV of -406,666,67 instead of -586,666.67 for Option1 (see status bar). But Option2 has also a much higher risk of total costs summing up to 920,000 in the worst case instead of -720,000 for Option1.

One has to make sure that the process of data gathering and estimating includes the worst cases data as well as data for the other cases (most likely, optimistic). Decision should then be made not only based on the most likely expectation but also on overall risk exposure.