Backward induction in procurement and mechanism design

About a month ago, I delivered my colleagues a training on backward induction, a method used in game theory to find Nash Equilibria in games that have pre-defined end points. I used materials and examples from the Open Yale course on game theory, since they were illustrative and easy to understand.

After the training, we had some discussion on when one can use backward induction, especially in real life, where we often have open-ended games. During the discussion, I was not able to provide any good example on when to use backward induction, but afterwards a great example came to my mind. Not surprisingly, this example was mechanism design.

Mechanism design is a sub-field of game theory focused on the study of creating and implementing mechanisms and incentives that align the interests of multiple agents. Earlier this year we had an external training on auction design, an application of mechanism design, where the goal is to design such an auction, that the buyer gets the best product for the lowest price. Obviously, auction design can be a powerful tool for any procurement professionals. Using auction design as an example of mechanism design, I will list some delimitations and pre-conditions for backward induction.

Backward induction in business transactions and relationships

Game theory and backward induction can be used in procurement to anticipate potential proposals and prepare counter-proposals, or prepare proposals to direct and distract the other party. Since a negotiation does not necessarily have a clear end from the start, using backward induction extensively can be very difficult, since the potential branches in the decision tree are not limited.

Another aspect in using backward induction is that all parties must be “playing the same game”. For example, if you are concentrated on the immediate outcome of the negotiation, while your opponent is looking at the long-term relationship between your companies, you might be going through very different decision trees with differing optimal outcomes for the immediate outcome. In addition, if the other party does not use backward induction but rather shoots from the hip, you cannot directly expect them to do choose optimally.

In procurement, our job is to create markets and create competition to leverage the market mechanism in buying the best quality we need for a good price. Therefore, mechanism design and game theory can be used to create such a game, a market, where the value to us can be maximized. The idea is to create an auction, where, using backward induction, the interests of the bidders and the buyer are aligned in such a way, that the buyer’s utility is maximized in the auction. Here it is important to make sure, that all the bidders are “playing the same game” and are deducing their actions backwards from the pre-defined end.

There is extensive research on auction and mechanism design. When we have set up the optimal auction design, we face another game, a meta-game, where we have to convince potential suppliers to participate in the bidding or auction, convince them that we are good customers and worth their business. Here we are again playing a long-term game without a pre-defined end, where the parties must convince each other of the benefits of doing business with each other and deliver on that promise. This relationship might also be the corresponding might also be aligned using mechanism design, but due to the scope it is a much more difficult task than a single auction.

Procurement – on savings and incentives

This summer our department at work published a global guideline on procurement. A colleague of mine did great work on managing a project where existing rules, guidelines, processes and tools were brought together, harmonized and supplemented. The result is a guideline that helps all company employees in procurement activities, whether it is about doing a demand and a supplier market analysis, preparing a tender, negotiating a contract or improving collaboration with existing suppliers.

In the following weeks after the publication, I gave trainings to colleagues from other departments on the most relevant policies and rules, available tools and the benefits of using the provided tools and processes. I was also contacted by some colleagues who were in need of procurement support, e.g. asking for help in initiating a supplier search. Being able to refer to the new guideline and the tools was a huge help for me. It reduced my workload immediately, and the people contacting me were happy to have such a clear guide on what to do and how to do it.

After these experiences, I was obviously convinced that soon we would have more of our colleagues buying more professionally and spending less. After all, procurement is something we do everyday in our personal lives: we buy food, bus tickets, plan buying a house, compare prices and so on. Still, companies have dedicated procurement departments. If procurement is so easy and intuitive, why doesn’t it work in larger companies the way it works in our private lives? Why do sensible people spend money less sensibly, when they are spending company money?

I think there are multiple factors here at play, at least two:

  • Lack of experience and information
  • Lack of incentives

Of these two, I find the latter to be the more convincing one, since it also contributes to the first one, but I’ll first briefly discuss the first point. When we do our everyday shopping, we have experience on what the goods usually cost, who are the market players and what is the quality we should expect. We are also usually quite able to define what we want. For example, I roughly know what a kilogram of apples costs in the nearby grocery stores so I know the market. I also know, what to expect from first class apples, so I am aware of the standard product quality. I can also define my preferences regarding apples, e.g. whether I like sweet or the more sour ones.

When you work for a company, you might often need external goods or services. If you do not purchase the same goods or services often, you lack market knowledge, including prices and the usual quality of the goods. For example, if you need a tax consultant, your knowledge on the competitive situation between consultants, their regular service spectrum and potential services might be unknown to you. Thus, even if you can define in detail what you want and what you are prepared to pay for it, you do not know if that is a good deal on the market. And should you be unable to define your needs and not know what kind of opportunities the market provides, you might be lured by a slick salesman in to buying some nearly related good or service that does not fulfill your needs or contains features you do not need at all. Continuing with the previous fruit analogy, you might not be able to define whether you want pears, prunes or apples and might end up leaving the farmer’s market with a cucumber in your bag.

A logical question is then, why we don’t dig out the necessary information and gather the experience. This is, of course, a matter of cost. The learning curve in the beginning is quite steep, and if you are, for example, a research engineer, the opportunity cost for learning the intricacies of the market for different sensors, for example, might be too high. Thus, we have dedicated procurement functions. But this leads to another question, namely, why do we see maverick buying, that is, people not using selected suppliers and the services of the centralized procurement function to gain market knowledge with smaller opportunity costs? Or why might it be that people do not use provided self-service tools that make tendering and offer comparison much easier, reducing the opportunity cost for the company significantly?

I get a penny, I lose a penny

Even if the most evident opportunity cost of getting competing offers from suppliers and doing solid offer comparisons is reduced, the added benefit for the single employee is not necessarily very high. In the case of the research engineer, getting to know the supplier market for sensors has the opportunity cost of doing research and development work that might lead to new products and income sources for the company. For the engineer there is an opportunity cost of potential bonus if the new products can be patented.

With a centralized procurement function the opportunity cost of the research engineer’s time may be reduced from the company’s perspective, if the engineer can quickly do an offer comparison and thus save a few thousand or more when buying the sensor. The generated savings would compensate the lost time in development work. However, for the engineer there might be nearly no benefit from doing this offer comparison. Assuming he is working on a new product and expecting to get additional remuneration from a potential patent, by doing offer comparisons he is not working towards his expected bonus for patent. Even if he reaches savings in excess of a thousand dollars, his payoff is practically zero, since those savings are either flowing in to the company’s profits or divided between all employees. In a very large company, dividing a thousand dollars between all employees, results in pennies per person.

On the other hand, our development engineer also does not suffer very much from using excess funds for buying the sensor. If he uses a couple of thousand dollars too much, this is again divided between all the company employees, incurring an effective loss of less than a dollar per person in a very large company. This leads us to a free-rider problem, where no one has the incentive to be the one to generate the savings and has therefore little reason not to spend excessively.

The problem here is that employees are not incentivized to use company money as if it were their own, since they usually do not see the effect of personally generated savings: it does not increase their salary or their bonus. Of course, excess saving, and lack of investments, today can endanger a company’s existence tomorrow, so we also a have a temporal problem. This kind of situation is also familiar in game theory, where certain games pose the question, under what circumstances people forego today’s added benefit for the anticipated future payoffs. Unsurprisingly, the present value of the future payoffs have to be large enough in comparison to the temptation created by the payoffs one would receive today. This is a question of time-preferences and the nominal values of the future payoffs.

In order to have employees use company money more effectively and efficiently, we need mechanisms that incentivize such behavior. Procurement professionals are sometimes paid based on achieved savings, but this might also be too naive a solution. For example, if the savings are measure against initial supplier offers, we can be quite sure that the procurement manager always comes with astronomically high initial offers, just to seal the deal with significant price reductions that conveniently guarantee him a good pay.

A mechanism to generate savings

The previously presented problem of aligning the incentives of the company and its employees is a principal-agent problem that can be solved with mechanism design. The principal, or the company, designs the mechanism, in this case the salary or the bonus structure of the employees, in such a way that each employee has the incentive to generate savings. Here it is assumed that the achieved savings exceed the potential opportunity costs for the company and for the single employee. After a quick Google-search I did not find any literature on this specific topic, but the principal-agent problem and mechanism design are broadly studied areas in game theory.

Concluding the previous discussion, a good mechanism that incentivizes all company employees to use the company money responsibly, should have at least the following features:

  • When a person generates savings, he profits from them more than other employees do per head;
  • Compensation for generated savings is at least as high as the employees opportunity cost, e.g. expected patent remuneration fees;
  • Compensation for generated savings is at least as high as the employees The net savings for the company are at least as high as the opportunity cost for the company, e.g. delay in new product development and market launch;
  • Generating long-term savings is more profitable for the individual than quick wins, thus arbitrary reduction of profitable investments is discouraged;
  • Savings are not measured against the initial supplier offer.

Signaling – Show that you can, show that you mean it

My previous post was about complete and incomplete information and about revealing your advantages to your opponents to gain an even bigger advantage. I finished with a short discussion on signaling: How to credibly differentiate yourself from others to gain a higher payoff?

In his course on game theory, Ben Polak represents a good example on signaling by using a simplified model of the job markets. Here I represent it, with possibly different figures, but the idea holds:

  • There are two types of workers only: good and bad
  • 10% of all workers are good, 90% are bad
  • a good worker produces 50 dollars worth of goods per day a bad worker produces only 20 dollars worth of goods
  • employers cannot tell the difference between a good and a bad worker before hiring them
  • the two types of workers are otherwise identical, just their output is different
  • this game lasts only one day, to keep the calculations simple

On average, an employee produces 23 dollars worth of goods, so the average salary level is also 23 dollars. Therefore, the bad workers earn slightly more than they produce, and the good workers a lot less than they produce: a good employee would earn 50 dollars if he could signal credibly to the employer that he is a good worker. Of course, a bad worker would also want to earn 50 dollars instead of 20 or 23. Thus we need something to differentiate between the two, we need a signal.

A signal that differentiates the good workers from bad workers, or any types from one another in general, has to be such that good workers will always give the signal and bad workers will never give the signal. For a signal to be credible, it’s costs obviously have to be such, that they reduce enough the net salary of a bad worker, but do not reduce too much the net salary of a good worker: this way the bad workers will not be willing to give the signal while the good workers will always give the signal.

As an example of a signal, Mr. Polak mentions the possibility of dancing on the table in a job interview and singing a song about how good an employee you would be. Such a signal is obviously costly, being humiliating at the very least, but it does not help differentiate the two types of workers. After all, it is equally humiliating for both types, so even if a good worker would have the incentive to give the signal, the bad worker would have the same incentive, in order to be identified as a good worker and thus receiving the salary of 50 dollars. Clearly not all costly signals can separate the worker types from one another.

Education as a signal

It turns out that education is a form of signaling and conversely, among other things education has a role as a signal giver at the job market. Let’s introduce a two-year MBA that either type of worker can take. The costs of tuition are the same for both and so are those for housing, transport and food. We might argue that the two types have different opportunity costs in taking an MBA instead of working, but both types would earn 23 dollars since we do not yet have a signal to separate them at the job market. So why does the good worker do the MBA and the bad worker doesn’t, as I am proposing? The difference in the costs is the effort, the mental work, hours of sitting in lectures doing homework and assignments. For the bad worker the required effort to finish the MBA degree is much higher than for the good worker. So much, that receiving an MBA would reduce his net salary below current levels, even below 20 dollars.

Of course, in this example the figures can be forced to be in such a relation to one other that the signaling works. E.g. if we make the total costs of an MBA, including the effort, to be 10 dollars per year for the good worker and 20 dollars per year for the bad worker, it is obvious that the good worker will do the MBA and receive a net salary of 30 dollars after being identified as a good worker and hired by an employer. Conversely, the bad worker will not take the MBA, since his net salary for doing an MBA and being identified as a good worker is 10 dollars, which is below the 20 dollars he would receive otherwise.  In addition, the employers have to believe that good workers, and good workers only, take an MBA. Otherwise the good workers might, regardless of their MBA, be identified as bad workers reducing their incentive to take an MBA.

Even if the figures in the above example are arbitrary, the main point is that the signaling mechanism has to be such that it will provide reliable signals and no type has the incentive to deviate. Thus, when creating the mechanism, the related figures actually have to be chosen in such a way that the signaling is reliable and adjusts the payoffs properly. In our MBA example a one-year MBA would not suffice, since a bad worker would do the MBA and receive a net salary of 30 dollars, instead of the 20 dollars he would receive otherwise. On the other hand, a one-year MBA could be made a lot harder and work-intensive, so that the per-year costs are increased and, again, only the good workers go for the education.

Signaling in other areas of life and work

Signaling is not only useful and used in employee-employer relationships. For example, buyer-seller interactions might also require, or at least benefit from, signaling.

For example in the case of used cars information imbalance between the buyer and the seller can lead to all goods in the market being of poor quality. In such a case, the potential sellers of higher quality products would have to be able to reliably signal this quality to the potential buyers. The buyers do not have to signal their preferences, since a buyer looking for higher quality products will buy one, if he gets more value from it, and nobody will pay for a good more than the value received from buying and possessing the good. Thus, we do not have the potential problem of customers looking for low quality products suddenly hoarding all the high quality products.

Another interesting realm where signaling can be applied is the one of procurement. At least larger companies often have a centralized procurement function that is responsible for managing the main suppliers, conducting supplier selection and awarding contracts. When looking for a supplier, the procurement function has to create competition between the candidates to find out the best one for the given quality and specifications. However, a potential supplier must invest resources in the bidding process without any certain revenue or supply contract. If the potential supplier thinks that the expected payoff from the contract is low, he will not put too much effort in to the bidding. This might lead to the customer company receiving only a few offers or the offers being of poor quality and difficult to compare against one another. To increase the number and quality of the offers, the customer company would have to signal, in a reliable way, that the potential contract has a guaranteed, maybe even high value. The way to do this is to commit, before the bidding starts, to awarding the contract to one bidder and to one bidder only, and beforehand abstaining from any cherry picking between offers. This of course requires that a large enough pool of bidders is invited and that the background research on them has been done, since procurement will have to commit to one of the invited bidders. Therefore, their capabilities have to match the requirements well enough, so that in principle any solution could be accepted.

As we see, tying your hands or incurring costs to show your commitment and capability are some ways to reliably signal who you are. Consequently signaling can help you leverage those advantages that might otherwise go unnoticed. Information is power, and shared information can be overpower.