George L. S. Shackle, “Three Versions of the $\phi$-Surface: Some Notes for a Comparison” — Summary

Shackle’s note compares two proposed revisions of his $\phi$-surface, by J. Mars and H. G. Johnson, in order to clarify what his own apparatus is meant to do. The essay is less a dispute over diagrammatic elegance than a defense of a division of theoretical labor: the $\phi$-surface locates standardized focus-values under uncertainty, while the gambler indifference-map expresses the decision-maker’s separate temperament toward possible gain and loss.

In my own scheme the $\phi$-surface serves purely to locate the standardized focus-values of a venture.

This sentence states the hinge of Shackle’s argument. The $\phi$-surface is not, for him, a complete preference machine. Once focus-gain and focus-loss have been standardized, they are plotted as monetary coordinates on an indifference-map whose contours need not be mathematically dictated by the surface that helped locate them. The point is to preserve a conceptual space between imagined possibility and personal valuation.

Mars’s variant is therefore the more consequential challenge. Shackle reads Mars as giving $\phi$ a positive-and-negative scale such that gains and losses can be added algebraically to yield a measure of attractiveness. If that operation is admitted, the indifference-map becomes unnecessary: the surface itself ranks ventures. What looks like simplification becomes, for Shackle, a loss of explanatory freedom.

By abandoning, or drastically circumscribing the role of, the gambler indifference-map, Mr Mars loses an essential ‘degree of freedom’ which my system possesses.

The geometrical discussion of straight lines, translated curves, and “crank-handle” forms is designed to expose this loss. Under Mars’s construction, indifference-curves are no longer independently interpretable expressions of a gambler’s attitude; they are generated mechanically from the $\phi$-surface. Shackle’s resistance is thus not merely formal. He wants to distinguish the imaginative salience of possible outcomes from the chooser’s willingness to accept particular combinations of hazard and opportunity.

No such rigid connection is implied by my own conception of the $\phi$-surface and the gambler indifference-map, and the latter is free to reflect a different aspect of the individual’s attitude to uncertainty from that reflected by the $\phi$-surface.

Shackle nevertheless concedes that Mars’s representation has visual attractions, especially in its smooth treatment of gains and losses. But he uses that concession to warn against allowing pictorial neatness to settle theoretical questions. His own preferred construction can represent a “subliminal” region: very small imagined gains or losses, when associated with extreme potential surprise, may fail to command the enterpriser’s attention. This is a psychological claim about salience, not just a feature of graph paper.

The final section turns to Johnson’s wooden model, which Shackle treats more sympathetically. Johnson’s version seems simpler and more symmetrical, and it may have pedagogical advantages. Yet Shackle remains cautious, especially where the model implies regions in which $\phi$ rises with potential surprise. Such possibilities are formally interesting, but Shackle doubts that they should displace the more flexible interpretive structure of his own scheme.

At a first glance, Mr Johnson’s version appears simpler than my own, and has a quasi-symmetry which, even apart from the pedagogic advantages claimed for the model, is undeniably attractive.

The note’s larger importance lies in its defense of non-reductionism in the representation of uncertainty. Shackle is not merely selecting among alternative surfaces; he is insisting that expectation, imagination, and preference cannot all be compressed into a single diagram without losing something essential. Mars’s model is criticized for making the $\phi$-surface too self-sufficient, while Johnson’s is welcomed as suggestive but not decisive. The central lesson is that mathematical elegance must not erase the separate psychological “degree of freedom” required to represent uncertainty as actually entertained and evaluated.