[lit-ideas] Re: Maximin Redux

  • From: Jlsperanza@xxxxxxx
  • To: lit-ideas@xxxxxxxxxxxxx
  • Date: Sun, 7 Jun 2009 13:46:21 EDT

In a message dated 6/7/2009 1:30:35 P.M. Eastern  Daylight Time,
wokshevs@xxxxxx writes:
I take it that few of us would make  love, or go fishing, or see the sunset
in NL) or walk the dog, or any  combination of the above, at a time when we
recognize that a loved one or a  stranger is in dire need of our aid. Many
our maxims are not consciously  formulated priot to action. All of our
actions, however, are  maxim-guided (oder?) and all maxims remain open to
Did  Asha borrow the term from Jack (Rawls) ?


I would guess so.

Asha Kasher is interested in the double force operating in the  principle
as a whole.


Notably for one of the categories -- the first: quantity (of information),
Grice provides two:

            1.  MAXI  -- be as informative as is required.
            2. MINI      do  not be more informative than is required.

Kasher thinks he can simplify things by appealing to a maximin  redux that
springs directly from rationality considerations. The same strategy
followed by Hintikka in PGRICE, "Philosophical Grounds of Rationality:  
Categories, Ends", but more in terms of game-theory. Stalnaker (MIT)  has
done even the same but along what he calls UTILITARIAN principles (and  so
closest to Jack Rawls).

For some quotes in the OED:

1949 Econometrica 17 184 The policy of ‘minimax regret’..may..be a
plausible description of the actual behaviour of some individuals. But
individuals of different temperaments might follow different policies: e.g.  
complacency’. 1954 D. BLACKWELL & M. A. GIRSHICK Theory Games &  Statist.
Decisions i. 27 0 is called a maximin strategy for player I. 1957 R. D.  LUCE
& H. RAIFFA Games & Decisions iv. 81 A strategy which dictates  choices c
and g on moves 2 and 4, respectively, cannot be maximin for this game.  1972
J. RAWLS Theory of Justice §26. 152 The maximin rule tells us to rank
alternatives by their worst possible outcomes. 1999 Nature 8 July 129/1 In a  
von Neumann was right. On the other hand, his own celebrated maximin
theorem reduces to a corollary of Nash's existence proof.

1953 Amer. Econ. Rev. 43 399 The player is maximizing the minimum pay off,
or ‘playing the maximin’. 1958 Engineering 21 Mar. 369/2 Player A will
thus  determine the smallest value in each row and choose the largest of these
minima,  the ‘maximin’. 1970 G. S. G. BEVERIDGE & R. S. SCHECHTER
Optimization xii.  656 This result represents the mini-max theorem of von 
Neumann and
Morgenstein  [sic] (1953), which states that A's expected gain is at least
P, and B's  expected loss is at worst L, where P = L... The value of P (and
L) lies between  the row maxi-min and the column mini-max.

[1928 J. VON NEUMANN in Mathematische Annalen C. 307 (heading) Beweis des
Satzes Max Min = Min Max.] 1949 Bell Syst. Techn. Jrnl. 28 704 A ‘max min’
type  of problem. 1955 Q. Jrnl. Econ. 69 103 Max-min Rule. Assume that
whatever  alternative is chosen, the worst possible outcome will ensue. 1961
Which? Nov.  277 (caption) Max-min speed switch. 1994 Proc. IEEE Internat. Conf.
Neural  Networks 7 4273 The FAM introduced by Kosko, which involves Max-Min
composition  units, is one of the most useful and important paradigms.

1950 Amer. Math. Monthly 57 511 The main theorem for such games is that 
there always exist optimal strategies for the two players, a strategy for the
one player that maximizes his minimum expectation, and a strategy for the
second  player that minimizes his maximum risk, the max-min equaling the
min-max. 1952  J. C. C. MCKINSEY Introd. Theory Games i. 19 Find the max min and
min max of the  two-finger Morra. 1991 Trans. Amer. Math. Soc. 325 492 The ‘
max-min’..is  achieved for Fi := span{1.., i} and u := i.

1962 Ecology 43 543 Differences between the psychometric dry bulb
thermometer and the *max-min thermometer inside the shelter are sometimes
relatively great. 1985 Gardening from ‘Which?’ Dec. 388 (heading) Max/Min
thermometer. 1990 Garden Answers Nov. 67/2 (advt.) A Max-Min thermometer is 
used  all
the year round, allowing you to keep an eye on the peaks of heat and depths
 of cold.

1917 Trans. Amer. Math. Soc. 18 240 If J0 is the value of J at a point P0
and if the inequality J < J0 -  where  is small and positive  defines more
than one region near the point P0, then P0 will be called a point  of
minimax. 1931 Amer. Jrnl. Math. 53 169 It has been proved..that (x,y)  
possesses a
minimax at each of the libration points L1, L2 and L3 and it  accordingly
follows that (x,y) must have an absolute minimum at L4 and  L5.

1958 Engineering 21 Mar. 369/2 Player B, on the other hand, will determine
the largest value in each column, and the smallest of these, the ‘minimax’
. 1962  Sci. Amer. Dec. 114/2 Both firms were guided by the principle of the
minimax,  choosing the best of the worst outcomes. 1968 Brit. Med. Bull. 24
232/2 This  would be the Bayesian Strategy of decision theory.., but other
strategies, such  as minimax or restricted Bayes, could also be used. 1970
G. S. G. BEVERIDGE  & R. S. SCHECHTER Optimization xii. 656 The value of P
(and L) lies between  the row maxi-min and the column mini-max. 1996 Economist
30 Mar. 96/2 At the  other extreme, a hypercautious government would prefer
‘minimax’the minimum of  maximums.

1917 G. D. BIRKHOFF in Proc. National Acad. Sci. U.S.A. 3 315 Another
method may be employed to find a large class of stable periodic orbits which I
call of minimax type. 1935 Amer. Jrnl. Math. 57 930 The minimum and minimax
methods..establish immediately the existence of an infinite number of
closed  periodic orbits. 1945 Ann. Math. 41 419 The critical limits, such as
absolute  minima, relative minima, minimax values, etc., are in general infinite
in  number.

1938 Amer. Jrnl. Math. 60 831 Let r be any integer such that H(r + 1)
assumes the maximum of H(m). The R(n)-block [r + 1, r + R(n)] will then be
termed a minimax n-covering in (a). 1940 Amer. Jrnl. Math. 62 237 Let sn denote
a value of s for which the maximum diameter of the sets Gs[te(n)] is a
minimum,..and let this minimax diameter be d(n). 1951 Ann. Math. Statistics 22
466 The minimax risk is (1 - a). 1990 Glasgow Math. Jrnl. 32 267 If G is a
finitely generated soluble-by-finite group of finite rank, then G is a
finite  extension of a soluble minimax group.

1952 Ann. Math. Statistics 23 587 For such weight functions the supremum of
 the risk over all    is infinite for every decision function, so  that
every decision function is minimax. 1970 R. B. ASH Basic Probability Theory
viii. 250 It is immediate from the definition of admissibility that an
admissible test with constant risk (i.e.,  = ) is minimax.

1947 A. WALD in Econometrica 15 283 An element a0 in A is said to be a
minimax strategy of player 1 if InfbK(a, b) takes its maximum value with
respect  to a for a = a0. 1949 Econometrica 17 230 Often Nature's strategy is
completely  unknown. In that case Wald suggests that the statistician play a
minimax  strategy: that is, the statistician should select that decision
procedure which  minimizes the maximum risk. 1957 R. D. LUCE & H. RAIFFA Games &
Decisions i. 2 Although Borel gave a clear statement of an important class
of  game theoretic problems and introduced the concepts of pure and mixed
strategies, von Neumann points out that he did not obtain one crucial
resultthe  minimax theoremwithout which no theory of games can be said to exist.
1978 M.  JEVONS Murder at Margin xiv. 138 If two people are charged with a
crime and are  separated by the police with no chance of their communicating,
there are certain  circumstances in which it would be rational for them to
confesseven if they did  not commit the crime. In economics this is called a
minimax strategy. 1993  Personal Computer World July 539/1 Many of the best
cognitive games programs  have a mini-max algorithm lurking somewhere or other
in their innards.

1957 Jrnl. Soc. Industr. & Appl. Math. 5 109 Our formulation of the
problem includes the possibility of minimaxing some measure of inaccuracy. 1957
Jrnl. Soc. Industr. & Appl. Math. 5 121 The loss L which is to be minimaxed
is the number of observations required before stopping. 1964 J. GOULD & W.
L. KOLB Dict. Social Sci. 573/2 The goal may be assumed to take the form of
maximizing (or, in game theory, minimaxing) the expected value, over some
time  interval, of a utility function. 1985 Ann. Statistics 13 1094 The
statistical  problem of minimaxing V(,f) over r ×  is meaningless unless the
least  informative member F0 has positive information.

1949 Amer. Econ. Rev. 39 205 Minimaxing the loss or, as I would prefer to
put it, maximining profits.

1953 Amer. Econ. Rev. 43 402 The Nash definition yields the same
saddle-point solution that is implied by the von Neumann-Morgenstern solution
concept (i.e., it corresponds to both players maximining). 1973 Philos. Q. 23  
He thinks that POPs would, in choosing their principles, ‘maximin’that is
to  say, choose which has the best worst outcome.

1949 Econometrica 17 183 Maximining of income..implies the ‘pessimistic’
expectation that nature will always tend to produce the situation most
adverse  to the operator and the ‘cautious’ unwillingness to take a chance about
it. 1973  Philos. Q. 23 184 This maximining approach often has
authoritarian results,  coercion being justified where this secures a higher 
1977 Jrnl. Econ.  Lit. 15 512 Hare writes that ‘this justifies only insurance,
not  maximining’.

1957 Rev. Econ. Stud. 25 2 In the two-person zero-sum game the process
called minimaxing ensures a payoff of a certain size. 1973 Nature 23 Feb. 507/1
 The fundamental mechanism underlying all this work [on machines designed
to play  games] has been a cycle of processes: lookahead, evaluation and
minimaxing. 1984  P. H. WINSTON Artificial Intelligence (ed. 2) iv. 117 Note
that the whole idea  of minimaxing rests on the translation of board goodness
into a single number,  the static value.

1968 Listener 4 Apr. 438/1 Just like that 50-50 combination of heads and
tails, the minimaxing peasant might pick a 40-60 combination of wheat and


J. L. Speranza
   Buenos Aires, Argentina

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