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Constrained Maximization

Published online by Cambridge University Press:  01 January 2020

Jordan Howard Sobel
Jordan Howard Sobel*
Affiliation:
Scarborough College, University of TorontoScarborough, ON, CanadaM1C 1A4
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Extract

This paper is about David Gauthier’s concept of constrained maximization. Attending to his most detailed and careful account, I try to say how constrained maximization works, and how it might be changed to work better. In section I, that detailed account is quoted along with amplifying passages. Difficulties of interpretation are explained in section II. An articulation, a spelling out, of Gauthier's account is offered in section III to deal with these difficulties. Next, in section IV, constrained maximization thus articulated is tested on several choice problems and shown to be seriously wanting. It appears that there are prisoners’ dilemmas in which constrained maximizers would not cooperate to mutual advantage, but would interact sub-optimally just as straight-maximizers would. ‘Coordination problems’ are described with which constrained maximizers might, especially if transparent to one another, not be able to cope–problems in which they might not be able to make up their minds to do anything at all. And I prove that there are prisoners’ dilemmas that, though possible for real agents and for straight maximizers, are not possible for constrained maximizers, so that agents’ internalising dispositions of constrained maximization could not be of help in connection with such possibly impending dilemmas. Taking constrained maximization as it stands, there are many problems for which it does not afford the ‘moral solutions’ with which Gauthier would have it replace Hobbesian political ones. After displaying these shortcomings of constrained maximization as presently designed, I sketch, in section V, possible revisions that would reduce them, stressing that these revisions would not be cost-free. Whether finishing the job of fixing up and making precise constrained maximization would be worth the considerable trouble it would involve lies beyond the issues taken up in this paper. So, of course, do substantive comparisons of constrained maximization, perfected and made precise, and straight maximization.

Type
Research Article
Information
Canadian Journal of Philosophy , Volume 21 , Issue 1 , March 1991 , pp. 25 - 51
Copyright
Copyright © The Authors 1991

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References

1 ‘Minimax relative concession’ signals Gauthier’s favoured solution-concept for bargaining problems. Its technical details, and its merits, are not important to present exercises. How it works, though important for these exercises, is explained in terms of a figure in section IV below.

2 Page references, unless otherwise indicated, are if unbracketed to David Gauthier, ‘Maximization Constrained: The Rationality of Cooperation,’ in Campbell, R. and Sowden, L. eds., Paradoxes of Rationality and Cooperation: Prisoner’s Dilemma and Newcomb’s Problem (Vancouver: University of British Columbia Press 1985) 75-93Google Scholar; and if bracketed to Gauthier, David Morals by Agreement (Oxford: Oxford University Press 1986).Google Scholar The material in Campbell and Sowden is there described as from ‘Ch. 6, sections 2 and 3 [of Morals by Agreement] with revisions made by David Gauthier’ (75). When words differ between these two texts, quotations are from Campbell and Sowden.

3 A fourth point, let me add, could be that qua strategy for making decisions, constrained maximization, CM, is not a ‘meta-strategy.’ It does not make an agent’s actions (even partly) depend directly and necessarily on (his views concerning) the strategies (dispositions for choice) and/or the (impending) actions of others. CM would make an agent’s actions directly and necessarily depend on several expected utility tests. One of these tests—the one in clause (ii)—is sensitive to his views concerning the actions and strategies of others. But this means only that CM would make an agent’s actions depend indirectly and contingently on views concerning other agents’ actions and strategies.

4 See Luce, R. Duncan and Raiffa, Howard Games and Decisions: Introduction and Critical Survey (New York: John Wiley & Sons 1957), 70 and 116.Google Scholar

5 The point here, I assume, is that an individual is not able to ensure that a joint strategy is enacted. If this is the point, then it needs to be qualified, for if a possible outcome has been selected in accordance with a joint strategy, and everyone else is doing their parts in it, then an individual can insure that it is enacted.

6 Cf. Gauthier: ‘ … where [that] natural harmony of interests [that makes possible an efficient competitive market] is lacking, we may seek a centralized mode of decision-making that will bring about an artificial harmony. In a democratic society, the choice of this mode becomes itself the object of interaction among individuals. We seek then a decision-making institution, within which individuals may be expected to interact in strategically rational ways, but which structures their interaction so that its product — the social decision — yields a fair and optimal state of affairs’ (‘Constituting Democracy,’ The Lindley Lecture, University of Kansas, April 7, 1989, 12; emphasis added).

7 Gauthier, David ‘Hobbes’ Social Contract,’ in Rogers, G.A.J. and Ryan, A. eds., Perspectives on Thomas Hobbes (Oxford: Oxford University Press 1988), 133Google Scholar

8 I count

as a prisoner’s dilemma if and only if a > b > c > d and a’ > b’ > c’ > d’, and each agent views the other agent’s actions as certainly causally independent of his own. In such a situation R1 and C1 are strongly dominant so that (R1,C1), and only this interaction, is a pure strategy equilibrium, and while it is not, each of the other pure strategy interactions is, an optimum relative to all pure strategy interactions. A prisoner’s dilemma is stretched if and only if 1/2(a + d)> b, and 1/2( a’ + d’) > b’. In a stretched prisoner’s dilemma, utilities of the joint mixed strategy [1/2(R2,C1);1/2(R1,C2)], exceed those of (R2,C2). In some stretched prisoner’s dilemmas, including the present one, 1/4(a + b + c +d)> b, and 1/4(a’ + b’ + c’ + d’) > b’: in these dilemmas utilities of the interaction of individual mixed strategies [(1/2 R1,1/2 R2); (1/2 C1,1/2 C2)], exceed those of (R2,C2), so that it is not an optimum relative to all strategy interactions.

9 Gauthier, David ‘Moral Artifice,’ Canadian Journal of Philosophy 18 (1988), 389CrossRefGoogle Scholar

10 Though they are sure they will not straight-maximize, I assume that they know that they could, that they are capable of straight-maximizing, that straight-maximizing is a possibility for them. And that is all that is required for well-defined (causal) expected utilities for their straight-maximizing.

11 This could happen if each, while not sure what they would do were they both to straight-maximize, considers it somewhat more likely than not (say 4/7 to 3/7 more likely than not) that each would perform the strategy that figures in this person’s most preferred outcome (R1 and C2 for Row and Column respectively). These views would lead, on the assumption that they both straight-maximize, to the following probabilities for the four possible outcomes:

Using these weights for the expected utilities of the outcomes yields the indicated expected utility of each agent for their both using individual strategies:

12/49(4)+16/49(0)+9/49(0)+12/49(3)=(48+36)49=12/7.

12 Similar problems come up for the version of Regan’s Coordinated Maximization Principle, COP (Regan, Donald Utilitarianism and Cooperation [Oxford: Oxford University Press 1980], 85CrossRefGoogle Scholar), that he claims is adequate to situations in which there are multiple best patterns of group behaviour: (COP-M). If there is a unique best pattern of group behaviour, do your part in it; otherwise, act in a way that would maximize the number of agents who would, given your action, be participating in a most widely participated-in best pattern of group behaviour (cf. 248; this is a paraphrase in which I have taken certain liberties, the main one of which is that where I have ‘a most,’ Regan would have ‘the most’). Regan argues that, assuming that actions of agents are causally independent (see 248, n. 5), if all agents are satisfying COP-M, then a best pattern of group behaviour has been realized. I have noted the importance of causal independence to his argument: see Sobel, Jordan Howard ‘Maximizing, Optimizing, and Prospering,’ Dialogue 27 (1988), 243,CrossRefGoogle Scholar n. 5.

I now add that notwithstanding Regan’s result, it is in some circumstances false that if each agent were to do what in these would satisfy COP-M, actions taken would constitute a best pattern of group behaviour. The difference signalled here by ‘all’ and ‘each … in the circumstances’ is the difference between what I have termed ‘universal projected’ and ‘universal actual’ conformities to rules (Sobel, ‘Maximizing, Optimizing, and Prospering,’ section 2.1; the same distinction is made in Sobel, Jordan Howard ‘Utilitarianism and Cooperation,’ Dialogue 24 [1985] 140,CrossRefGoogle Scholar and ‘Everyone’s Conforming to a Rule,’ Philosophical Studies 48 [1985) 376-7 and 382-3). Here is a case to this point:

Suppose acts are independent and that Row will do R1, and Column will do C2. Then COP-M requires R2 of Row, and C1 of Column, which actions do not constitute a best pattern of group behaviour.

Regan observes that COP-M ‘directs each agent to consider the other agent’s behaviour in deciding [in which best pattern of group behaviour to participate]’ (249). It is clear that this dimension of CO P-M threatens, in cases such as the one before us, to incapacitate agents who know one another to be adherents of COP-M.

13 Gauthier, David ‘Reason and Maximization,’ Canadian Journal of Philosophy 18 (1988),Google Scholar 428 with assistance from 424

14 Ibid., 429

15 Sobel, Jordan Howard Interaction Problems for Utility Maximizers,’ Canadian Journal of Philosophy 4 (1975), 687CrossRefGoogle Scholar, n. 7

16 Cf.: ‘Suppose you and I are CMs and recognize each other to be so in a PD …. How will I (you) choose as a CM? I am disposed to cooperate if you will. You are disposed to cooperate if I will. But that doesn’t settle what each of us will do …. We are both ready to cooperate and indeed we will, should each expect our actual choices to be cooperative. But how are we any closer to a decision?’ (Richmond Campbell, ‘Gauthier’s Theory of Morals by Agreement,’ The Philosophy Quarterly 38 [1988], 351)

17 I argue in my (unpublished) manuscript, ‘Two Arguments To Do with Straight and Constrained Maximization’ (July, 1990), along lines very similar to those used above for Case 3, that standard prisoners’ dilemmas,

are not possible for oonstrained maximizers (as identified in our Primary Quotation) who consider somewhat more likely than not that the other party will do the ‘right’ thing, -C, and not confess.

18 The lines explored are conservative. Revisions considered do not include changes that, while attempting to deal with these problems, made constrained maximization into a complex ‘meta-strategy.’ For discussion of simple meta-strategies that are somewhat similar in spirit to Gauthier’s constrained maximization, see Danielson, Peter The Visible Hand of Morality,’ Canadian Journal of Philosophy 18 (1988) 357-84CrossRefGoogle Scholar. Also of interest is Howard, J.V. ‘Cooperation in the Prisoner’s Dilemma,’ Theory and Decision 24 (1988) 203-13CrossRefGoogle Scholar.

19 The general condition underlying the impossibility of this indeterminacy, in terms of the Figure in Section IV (1), is that each point in the triangle is for expected utilities for at least but not necessarily exactly one joint strategy.

20 Had Row’s ‘prior’ probabilities made C1 and C2 equally likely, her ‘prior’ expected utility for basing her action on [1 /2(R1,C2), 1 /2(R2,C1)] would be exactly 2.25. Had she not only considered C2 somewhat more likely, but been sure of it, her ‘prior’ expected utility for basing her action on that strategy would be 7/2, and so considerably greater than 2.25.

21 This recommendation and the coming revision pursuant to it respond to the observation that ‘prospects for cooperation should not be adversely affected by the absence of joint randomizing procedures where there is mutual advantage to be gained … without them’ (David Gauthier, private correspondence).

22 Part of the apparent intent of constrained maximization in an early formulation was that constrained maximizers should be kinds of agreement-keepers. Perhaps this should be taken as implicit in the current formulation, designed as it is to figure in a theory of morals by agrmnent. As noted in section IV(2c), constrained maximization was once identified with a ‘policy, which requires individual utility-maximization [when one acts “independently”] … and agreed optimization [when one acts “interdependently”, in a manner agreed by all]’ Gauthier, ‘Reason and Maximization,’ 428 and 424 with emphasis added). (Also consider the occurrence of the phrase ‘in receiving the voluntary agreement of all’ in the post Morals by Agreement passage quoted in section IV (1b) above.)

23 Regan’s ‘cooperative utilitarianism’— which in summary statement would have ‘each agent … co-operate, with whoever else is co-operating, in the production of the best consequences possible given the behaviour of non-co-operators’ (Regan, 124) — is to involve its adherent safely in ‘coordinated optimizations’ without the aid of agreements and without dependency on authoritative identifications of optima in which agents are to congregate. The restriction of constrained maximization to cooperative arrangements that have been authoritatively selected, and the empowering of constrained maximizers to arrange for such selections by explicit agreements, would distance constrained maximization from cooperative utilitarianism in two ways in which I have thought that they were supposed to be close.

24 In ‘Two Arguments to Do With Straight and Constrained Maximization’ (unpublished ms, July, 1990), I criticize the ‘letter’ of an argument Gauthier uses to show that, given a choice, ideal transparent agents would take on the disposition of constrained maximization rather than of straight. And I repair to nearly full force another argument he rejects that would show the converse.

25 I am grateful to Richmond Campbell, Włodzimierz Rabinowicz, Willa Freeman Sobel, and Peter Vallentyne for valuable comments made on drafts of this paper, and to David Gauthier for providing many good things to think about, and for helpful criticisms.