Two Alternatives for Fair Optimal Solutions

“A Beautiful Mind” was a popular movie released in 2001 documenting the true story of John Forbes Nash, Jr., a mathematician who developed several groundbreaking concepts in economics. The importance of Nash’s work has been recognized by many honors: the von Neumann Prize, fellowship in the Econometric Society and the American Academy of Arts and Sciences, membership in the US National Academy of Sciences, and culmination in the 1994 Nobel Prize in Economics.

How Smartsettle relates to all this is becoming a frequently asked question. Here is the answer.

**The Bargaining Problem**

Negotiation is hard work. Negotiators frequently fail to reach agreement even when a good solution was possible, and when they do, they often leave significant value on the table. Game theorists, mathematicians, economists and systems analysts have been wrestling with this for a long time. The problem is how to help negotiators achieve both fair and efficient resolutions.

**Maximize the Utility Product**

In 1949, while studying for his doctorate at Princeton, John Nash developed a model showing that competitive behavior among decision makers leads to a non-optimal equilibrium (now known as the Nash Equilibrium). This radical idea challenges the classical economic theory of Adam Smith, where free competition leads to best-possible results, and is in contrast to classical Darwinian theory, where natural selection leads to improvement in the species. The dynamics of unregulated competition can actually be disastrous (Milnor, 1998).

In 1950, Nash wrote his Econometrica paper entitled “The Bargaining Problem”. In this paper, Nash showed that a unique optimal solution could be found by maximizing the product of the utilities for cooperative negotiators. Nash theorized that “we idealize the bargaining problem by assuming that the two individuals are highly rational, that each can accurately compare his desires for various things, that they are equal in bargaining skill, and that each has full knowledge of the tastes and preferences of the other.” These were also the arguments for this solution being the fairest possible outcome.

Over the years, Nash’s seemingly simple ideas have led to fundamental changes in economics and political science (Milnor, 1998). Estimates of benefits to business are already in the billions of dollars (1). This work also has implications for world peace. If superpowers could find a way of cooperating, then this would lead to a reduction in weapons and to cost savings for all sides (Singh, 1998). Unfortunately, Nash’s theories stop short of prescribing how to cooperate, which is an even more complicated problem (Milnor, 1998).

Where Nash and Thiessen part ways is the requirement of parties to fully know each other’s preferences. Realizing that real-world negotiators would not cooperate in that way, Thiessen specified a secure neutral site to fulfill the knowledge requirement.

**Maximize the Minimum Gain**

While studying for his doctorate in engineering at Cornell University, Ernest Thiessen was working on a challenge given to him by his major advisor, Professor D. Pete Loucks. He needed a rule for fairly distributing benefits when generating an optimal solution relative to an existing tentative agreement among any number of negotiators. The mathematical framework for Thiessen’s solution was inspired also by Howard Raiffa’s work (2) (3). Since Thiessen’s goal was to develop a computer program to demonstrate the results of his research, an important criterion was that the methods would perform well in practical implementation. Thiessen came up with a rule called “Maximize the Minimum Gain”, which turned out to be very well behaved using proven mixed-integer linear optimization techniques. Descriptions of these algorithms were first published in 1992 (4).

As with Nash’s theories, Thiessen’s method of generating optimal solutions also required rationality and well-represented preferences. Where Nash and Thiessen part ways is the requirement of parties to fully know each other’s preferences. Realizing that real-world negotiators would not cooperate in that way, Thiessen specified a secure neutral site to fulfill the knowledge requirement. Thiessen’s methods, now recognized as a patented invention (5), are implemented in the Smartsettle eNegotiation System (6).

**Conclusion**

In “Lectures on Negotiation Analysis” (7), Professor Emeritus Howard Raiffa (Harvard), a respected pioneer in the field of decision analysis, argues that the “Maximize the Minimum Gain” rule (8) may actually be better than the “Maximize the Utility Product” rule for producing fair optimal solutions (9). However, Raiffa also concedes that which of these algorithms is used makes very little difference in practice. They both produce solutions that tend to be close together in most non-artificial cases. In the last paragraph of his book, Raiffa identifies Thiessen along with several others as “doing cutting-edge research on the development of software for computer-aided mediation and conflict resolution.”

(1) An example of the application of Nash’s noncompetitive game theory to business negotiations was in the auctioning of bandwidths for wireless communications in 1994. This application yielded an estimated $10 billion in revenue for the US Federal government (McAfee, 1998) (Milnor, 1998).

(2) 1982, Howard Raiffa, The Art and Science of Negotiation, Belknap Press of Harvard University Press, 1982.

(3) 1985, Howard Raiffa, “Post Settlement Settlements”, 1 Neg J 9.

(4) 1992, Thiessen, E.M., and D.P. Loucks, “Computer-Assisted Negotiation of Multi-objective Water Resources Conflicts,” Water Resources Bulletin, American Water Resources Association, 28(1), 163-177, February.

(5) 1996, US Patent 5,495,412 (ICANS)

(6) The methods discussed in this article presume that negotiators have already reached a tentative agreement or have otherwise determined criteria for a fair solution. Recent research (10) shows that conventional negotiations often leave negotiators exhausted by the time they reach agreement. In the year 2000, Ernest Thiessen and Ian Upright invented another method called “Multivariate Blind Bidding”, with which negotiators using Smartsettle can reach a tentative agreement relatively quickly. This method also tends to produce initial solutions close to the efficiency frontier, making it even easier for negotiators to reach an optimal solution.

(7) 1996, Howard Raiffa, Lectures on Negotiation Analysis. Program on Negotiation at Harvard Law School.

(8) Raiffa identifies this method as falling into a more general category called “maximin”, which is short for “maximize the minimum”. The name “maximin” has also been applied to many other situations in which the idea is to choose the course of action that is likely to cause the least harm. Perhaps the most famous of these is Rawls’ Maximin Rule. In “A Theory of Justice (1971)”, John Rawls tackles the topics of social justice, equality, and freedom. Opposing utilitarianism, which would maximize overall good without regard to how benefits are distributed, Rawls came up with a rule called the “difference principle” or “maximize the minimum utility”. Rawls’ rule guarantees benefits in society first for those most disadvantaged. Kemerling (2001) describes Rawls’ rule as “maximization of the minimum gain”, a rational guide for social decision-making.

(9) 1999, Teich, Jeffrey, Hannele Wallenius and Jyrki Wallenius, Multiple-issue auction and market algorithms for the world wide web. Decision Support Systems (26) 49-66.

(10) 1999, Shell, G. Richard, Bargaining for Advantage: Negotiation Strategies for Reasonable People. New York: Penguin, 1999. ISBN 0 14 02.8191 6 paper.