Nash Equilibrium is Not a Theory of Bargaining

Eric Beinhocker’s Origins of Wealth continues to be my ‘in-between-writing’ reading and it is full of fascinating insights into complex adaptive system thinking, which he believes, and I agree, promises to be a richer paradigm for research than the equilibrium research paradigm that has dominated economics for the last few generations and is almost lost in abstraction, increasingly divorced from reality. Hence, I continue to recommend that you read his book.

However, as with Beinhocker’s account of Adam Smith’s philosophy and political economy, he continues to demonstrate gaps in his understanding of other areas of expertise in subjects with which I am familiar. The most recent is in his account of John Nash’s solution to the ‘Bargaining Problem’ (Econometrica, 1950) (Origins of Wealth, pp 267-8).

Beinhocker says that this problem ‘stumped economists for generations’, which it did, within the confines of neo-classical equilibrium. Unlike, the determination of prices in a competitive market, the price in bargaining situations (such as trading between two nations, monopoly versus monopsony, and negotiations between employers and trade unions), were regarded as indeterminate, or at best a fuzzy grey area – bargained outcomes fell in a range of possible prices according to the bargaining skills of the buyer and seller.

Nash did not resolve the bargaining problem, brilliant as his 1950 paper was and remains. He engineered the outcome of bargaining– not the process of bargaining – into neo-classical equilibrium. It is pure homo economicus (Beinhocker does not appear to have noticed this at all). He has bought right into the notion that Nash demonstrated the pay-offs from co-operation – yes, but only by Nash assuming they were not bargaining at all! He writes:

‘Nash’s elegant solution was to say how two or more bargainers split up the gains from exchange depends on how much each values the benefits of the deal, and what the parties’ alternatives are. Each looks for his or her best deal assuming everyone else is looking for the best deal too, and the trade is made at the point at which no one has any incentive to change position, given the actions of the other. … Both walk away better off that they would have been had they not traded at all, thus capturing the on-zero sum gains of co-operation.’ (pp 267-8)

Now, Adam Smith stated the bargaining solution in 1776 and his work on this problem lay ignored by economists until the 1970s. Edgeworth stated a Nash solution in Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences, (London 1881) in the exchange between two bargainers at the frontier of their preference curves in what became known as his ‘box diagram’; neither can move to improve his position without making the other worse off (also expressed as a Pareto Optimum).

Smith stated the conditional proposition (Wealth of Nations, Book 1, chapter ii): ‘give me this what I want and you shall have that which you want’, which is the meaning of all bargain propositions.

Nash, however, assumed: Highly rational bargainers, who can accurately compare each other’s desires; they have equal bargaining skills (by assuming them away); who have full knowledge of each other’s tastes and preferences (preventing bluffing and ploys) and they desire to maximise their gains – the difference between what they ‘give up’ compared to what they ‘get’ in exchange.

In short, the same unrealistic assumptions of neo-classical economics, which Beinhocker exposes as a main weakness of neo-classical modelling in the early chapters of Origins of Wealth, and which complexity theory is seeking to replace in line with modern developments in maths.

Smithian bargainers were advised in 1776 to trade that which is cheap for you to give in order to get back that which you value more. Nash’s example of two bargainers ‘Bill’ and ‘Jack’ exchanging toys shows how when they do this, under his assumptions that remove the process of bargaining, they will maximise the product of their net gains in utility. But he does not model how they do this, or, more realistically, how they would approach the Nash/Pareto/
Edgeworth equilibrium, in practice. Like the mathematics of General Equilibrium, it is evidence of an ‘elegant solution’ (even a ‘beautiful mind’), but did not resolve the bargaining problem, because between the opportunity to improve one’s utility and enjoying the Nash ‘solution’ there is the awkward process by which bargainers in the real world attempt to reach it.

In the theory of perfect competition bargainers have no choice but to accept the competitive price, or do without. In bargaining, they have choices. The ‘rational’ choice is to accept what somebody offers for something you value less than what you get in exchange by giving up what they want. This means not worrying about ‘leaving money on the table’, or which of you ‘gains most’, and all the other silly notions that unthinking people bring to the bargaining process in practice (including the Negotiation paradigm taught at Harvard).

If somebody offers you something that you value, and you value more than what you give up in return (Smith’s conditional proposition) then your best inclination is to accept the deal. If you think you can get back more for what you give – thus, perhaps, walking into the ‘greedy’ not the ‘needy’ ‘trap’ – but by mediating your interests and taking account of, and ‘addressing their interests’, not just your own, the problem of ‘selfishness’ is overcome. As people are the best judge of their interests, offers made that make you better off, also by reasonable inference, make them better off too in their minds You both move towards, but maybe not onto, a Nash/Pareto/Edgeworth frontier.

The Hollywood scriptwriter in ‘Beautiful Mind’ (the film, which was also a misleading account of John Nash’s life) has Nash saying he has ‘proven’ Smith wrong. That depends on which Adam Smith we are talking about: Chicago Smith or Kirkcaldy Smith? Kirkcaldy Smith was interested in bargaining as a process and how the solutions reached by practical people made them better off – it enabled them to become dependent, safely, on others and to improve their living standards (which means increasing their dependence).

Just as it did not require perfect liberty for progress towards opulence to take place, it does not need everybody bargaining soley to maximise the product of their net gains. If they do, fine; in its absence, also fine. In evolution, very small gains also count!