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Group Optimization – An Application of the Nash Equilibrium

Michael Lieberman

February 1, 2019

BLOG_Group Optimization_Nash Equilibrium

The Nash Equilibrium—see A Beautiful Mind—in economics and game theory is defined as a stable state of a system involving multiple participants, where no one can gain by a change of strategy if the strategies of the others remain unchanged. More simply, it is a maximized state where no other players will agree if one player changes her position. In terms of economics or business, it is the most equitable, though not always the most obvious, solution to a multi-party conflict.

Applied behavioral economics gives us an example of how illogical situations can be solved Thinking Fast and Slow-Kahnemanmathematically by use of Game Theory to reach the Nash Equilibrum. Behavioral economics is a “field that studies the effects of psychological, social, cognitive, and emotional factors on the economic decisions of individuals and institutions.” Its most well-known use describes our nearly universal 'Loss aversion': our tendency to strongly prefer avoiding losses compared to acquiring gains. Most studies suggest that losses are twice as powerful, psychologically, as gains. Loss aversion was first demonstrated by Amos Tversky and Daniel Kahneman, and was part of their work winning a Nobel prize. Kahnesman has recently spawned a new industry, System 1 research, using concepts popularized in his excellent book, Thinking Fast and Slow.

Applied behavioral economics can address group optimization among competing entities. ‘Nobody wins’ is often a mantra for successful negotiations. Loss aversion is usually the villain, as it is not obvious that surrendering on some points can be to one’s benefit. Of course, that is not always the case.

In this piece I give an example of a “group optimization” problem. It’s a technical approach to solving a tough problem, where everyone has to be satisfied enough or the whole project won’t go forward. It is a practical example of the utility of finding a Nash Equilibrium.

Fair Division of Costs

A tough business problem is how to share costs and benefits from a joint investment or project; that is, how to cooperate, among different companies, say, where the project can’t proceed without cooperation from all. Take a popular example called the ‘airport problem’. Four airlines are sharing an airstrip, each with different needs for the facility. For example, the airline with the largest planes (let’s call it Airline D) needs a longer strip. Airlines with smaller planes (let’s call the smallest Airline A) can use a longer strip, though it is not required. Thus, they do not want to pay the additional costs of maintaining a longer runway. And so on: each airline has different ideas of what they need and are willing to pay for. Yet, without agreement all will lose. How can we determine how the cost should be divided among airlines, so that acceptance is likely?

I am confident that Airline D executives would propose the cost of building and maintaining the airstrip should be equally spread between the four airlines. After all, all of them are using the facility. I am sure Airline A would disagree, chafing at the shared cost of too much runway. And Airline B and C have opinions too! A good algorithm can break this impasse.

We can compute the Nash Equilibrium using algorithms which are now available, open-source, from the R stat package named ‘GameTheory’. Below is an example of an input line:

> BUILDINGCOST <- c(8000,11000,13000,18000)

> AIRLINES <- c("Airline A","Airline B","Airline C","Airline D")

Building cost is related to the length of runway required by each airline, which are named in the second line. The Shapley Values are the most equitable solution calculated by the software.

The proposed solution for our airport problem is:

  • Divide the cost of providing the minimum level of required facility for the smallest type of aircraft equally among the number of landings of all aircraft
  • Divide the incremental cost of providing the minimum level of the required facility for the second smallest type of aircraft (above the cost of the smallest type) equally among the number of landings of all but the smallest type of aircraft.
  • Continue until finally the incremental cost of the largest type of aircraft is divided equally among the number of landings made by the largest aircraft type.

Table 1_An Application of the Nash Equilibrium

Examining the table, we see that the startup of adding Airline A is high, but Airline D, which is the last to be added, has the largest capacity. Thus our Shapley Value column has calculated how much each airline should pay for the facility. Airline D (or another) might not like it, but this is the most equitable solution, mathematically, and provides an excellent starting (or ending) point for negotiations.

Conclusion

It is valuable to know that there can be a straightforward way to “game theory” a “win-win-win” solution to a tough negotiation. This methodology extends not only to competing claims, but to power calculations, most equitable payout of bonuses, and a host of other business challenges. As computing power expands, the combination of agile data expertise and strategic outcomes becomes more powerful and accessible.


Editor's note:
Michael Lieberman is founder and president of Multivariate Solutions, a New York data science and strategy firm.


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About the Author

Michael Lieberman Guest author Michael D. Lieberman has more than 25 years of experience as a researcher and statistician in the marketing and advertising research fields. He has worked extensively with clients in financial services, information technology, food service, telecommunications, financial services, political polling, public relations, and advertising testing fields. He founded Multivariate Solutions in 1998 and now works with an international clientele including advertising firms, political strategy groups, and full service market research companies. Mr. Lieberman has taught at City University of New York and the University of Georgia as an adjunct professor of statistics and market research. He holds a B.A. degree in Mathematics for Rutgers University, an M.A. degree in International Affairs from George Washington University, and an M.S. degree in Statistics from Rutgers University.