# On Numbers and Games / Cooling

Sub-page of OnNumbersAndGames

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.

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.

On Numbers and Games / Cooling last edited by RobertPauli on January 30, 2019 - 11:08