On Numbers and Games / Cooling

Quote (first edition, 1976, chapter 9, p. 103)

So we shall define a new game ``G_t`` (``G`` cooled by ``t``) by charging each player a fee of ``t`` every time he makes a move, until the value becomes a number. A formal definition is complicated slightly by the need to detect when this has taken place.

Definition. If ``G`` is a short game, and ``t`` a real number ``>= 0``, then we define the cooled game ``G_t`` by the formula

``G_t = { {:G^L:}_t - t | {:G^R:}_t + t }``,

unless possibly this formula defines a number (which it will for all sufficiently large ``t`` ). For the smallest values of ``t`` for which this happens, the number turns out to be constant (that is, independent of ``t`` ), and we define ``G_t`` to be this constant number for all larger ``t``.

John’s definition is very delicate:

• the unless clause revokes what just seemed to be defined,
• the plural in “smallest values of ``t``” is very important (a single letter!) and only makes sense if the formula partially becomes constant, and
• "for all larger ``t``" seems to dangle in the air but apparently means those ``t`` whose ``G_t`` was a number (before it possibly was changed into another number ;-).

Let’s disentangle his definition. ONAG indeed has a distinction between ``G_t``, the one with the unless exception, and ``G(t)``, the one without, but unfortunately does not make use of it. IMHO the definition would benefit:

``G_t = G(t)`` for ``0 <= t <= r`` and ``x`` for ``r < t``, where

``G = { a, b, ldots | p, q, ldots }``

``G(t) = { a_t - t, b_t - t, ldots | p_t + t, q_t + t, ldots }``

``r =" the smallest "t >= 0`` for which ``G(t')`` converges to a number ``y(r)`` if ``t' > t`` converges to ``t``

``x = G(r)`` if that is a number and ``y(r)`` otherwise.

For examples see Cooling / Examples.

— Robert Pauli

Patrick Traill: Robert, is not the significance of the plural “values” that there need not be a single smallest value for which ``G_t`` is a number? As in the example ``G = { 4 | 1 }`` (on the same page), for which

“ ``G(t) = 2½" for "1½ < t ⩽ 2`` ” (Note the strict inequality ``1½ < t``)

You evidently realise this, since you formulate in terms of convergence, but I feel that that is itself a sophisticated concept, especially as it requires a topology on short games.
I also do not really see “for all larger ``t``” as dangling, but as belonging fairly naturally to “and we define ``G_t`` to be this constant number”; what I find less clear at first sight is the referent of “larger”, which is of course “the smallest values of ``t`` for which this happens”.
It is also notable that on the next page, in the proof of Theorem 60, he says “For the moment, we are continuing to assume that ``G_t`` is well-defined.” (i.e. that the assertion is true – but ¿does he mean the existence of a ``t`` making ``G(t)`` a number or that the number is constant? ¡presumably the latter, as the former is relatively obvious! – suggesting that it is to be proved later, as indeed he says it is just after the proof of Theorem 61.
I agree in wanting to reformulate somewhat, but I am not sure how, I just feel that you have not got there yet. ¡Maybe by the time one feels able to, his definition feels natural anyway!

Robert Pauli: Here’s my (late) answer, Patrick.

• Concerning the plural, I already had prepared an erratum pushing John’s nose into his very example until I realized that I missed a single letter. :-)
• Concerning the convergence, I see no problem since at the point it is asked for, ``G(t')`` is not only a number, it’s a constant number.
• Concerning the dangling "larger ``t``", the question could be, larger than what? Let’s look at his example (``G = {4|1}``). The smallest values of ``t`` for which ``G(t)`` becomes a constant number are those ``> 1.5`` and ``<= 2``. Now "we define ``G_t`` to be this constant number for all larger ``t``" could be very naturally taken as those ``t`` that are larger than those just mentioned, i.e. ``t > 2`` instead of the correct ``t > 1.5``. Do I have a point? I’m at the verge of an erratum again. :-)
  Wrong. I don't have a point because to take this number for all t > 2 in the example is the same as to take it for all t > 1.5 since for 1.5 < t <= 2 it already is taken.
• BTW, Patrick, as far as I know, English works without ¿ and ¡, not?
  • PJT: It certainly works without them, and that is conventional – I just somewhat idiosyncratically think that now and then it works even better with them as delimiters!
• Concerning well-definedness, I think it’s both, the claimed but not yet proven existence of "the smallest values of ``t`` for which this happens" and that the numbers those ``t`` yield actually are one number.
• Concerning my reformulation, I felt John’s formulation to be a little airy, at least to me, so dummy me tried to be as explicit and down to earth as possible. To indeed use ``G(t)`` is without question to me.
• Concerning your reformulation ( . . . that is gone? It wasn’t bad, Patrick, I used it below!)
  • "there is a number ``t``" should be "there is a smallest number ``r``", sparing ``t``, avoiding subscripts (despite being able now!), and fitting to the example page.
  • The constant claim should be:
    For every game ``G`` there is a largest number ``s`` such that ``r < t <= s`` implies ``G(t) = G(s)``.
  • BTW, Patrick, I feel less " in the source, if any, is better. The formulas are the exception, not the text. Problems with the backtick? I get two if I hold down shift and hit backtick twice (the first hit shows nothing, however). So, hit it four times and insert.
    • PJT: I agree that " for what {:…:} does is undesirable, in other cases I find it a matter of taste. I can insert backticks easily.
  • Your last definition then becomes:
    ``G_t = G(t)`` for ``t <= r`` and ``G_t = G(s)`` for ``t > r``.
  • Since I like your way to put it, let me apply it:

  ``G_t = G(t)`` for ``0 <= t <= r`` and ``G(s)`` for ``r < t``, where

  ``G = { a, b, ldots | p, q, ldots }``

  ``G(t) = { a_t - t, b_t - t, ldots | p_t + t, q_t + t, ldots }``

  ``r = min {`` number ``r' >= 0: AA t > r':`` ``G(t)`` is a number ``}``

  if ``G(r)`` is a number

  then ``s = max {`` number ``s' >= r: AA t >= r:`` ``t <= s' => G(t) = G(s')`` ``}``

  else ``s = max {`` number ``s' > r: AA t > r:`` ``t <= s' => G(t) = G(s') `` ``}``

  The existence of ``r`` and ``s`` for each game ``G`` is yet to be shown,

  and ``r`` is called the temperature of ``G``.

Patrick Traill: It is now too late at night for me to check all that, but perhaps I shall have time tomorrow.

Robert Pauli: I forgot the tricky case where ``r`` throws a number but no ``t > r`` throws this number too. Therefore now the ugly if-then-else since, keeping up your idea, I wanted to have ``s`` always be the end of the first interval that throws numbers (possibly ``r`` alone). Maybe you find a more elegant way.