Betting Strategy

Date: 05/02/2003 at 13:58:42
From: Don
Subject: Betting strategy

Dear Dr. Math:

Suppose 2 teams play a series of up to 7 games in which the first team
to win 4 games wins the series, and then no more games are played. 
Suppose that you want to bet on each individual game in such a way
that when the series ends you will be ahead by exactly $100 if your
team wins the series, or behind by exactly $100 if your team loses the
series, no matter how many games it takes.  How much would you bet on
the first game?

My answer was that I would bet $100 on either team in the first game,
then none for the rest of the series. By this way, I am ensured to win
or lose exactly $100. Is my answer too primitive, as I am told? I
guess there is some tricky word-game in the statement of the problem.

Thanks a lot.
Don

Date: 05/02/2003 at 16:11:03
From: Doctor Douglas
Subject: Re: Betting strategy

Hi Don,

Thanks for writing to the Math Forum. Your answer doesn't quite work, 
because suppose your team loses the first game but wins the next four 
and the series. This earns you bragging rights but you are out $100.

At first I thought that this problem would have many possible 
solutions, but then in working it out carefully it seems that there is 
only one strategy that works to achieve a final gain or loss of 
exactly $100 no matter what happens in the series.

Here's the reasoning. Suppose it is just before the seventh game, and 
the series is tied 3-3. Can you be out of pocket? No; otherwise it is 
impossible to achieve +$100 if your team wins, and -$100 if your team 
loses game 7. Thus, at the beginning of game 7, you must be even (in 
money terms), and must bet $100.

Now consider that it is just before game 6, and let's imagine that the 
game score is 3-2. To calculate how much money must be in your pocket 
now, we imagine the two outcomes of game six. Either your team wins 
(4-2) and you are at +$100, or your team loses, and the series is tied 
3-3, and your wallet is standing at $0. As game 6 starts, you must be 
halfway between these two possible outcomes (you have made $50 so 
far), and your bet on this game must be for $50.

We continue to work backward in this way, seeing what the two outcomes 
of the game could be, and demand that just before the game starts our 
wallet be exactly halfway between the two outcomes:

      G7     G6      G5     G4       G3     G2        G1
---------------------------------------------------------------
4-?
+$100                        3-0,+$87.50
                     3-1,+$75        2-0,+$62.50
             3-2,+$50        2-1,+$37.50    1-0,+$31.25
      3-3,$0         2-2,$0          1-1,$0           0-0,$0
             2-3,-$50       1-2,-$37.50     0-1,-$31.25
                     1-3,-$75        0-2,-$62.50
?-4                         0-3,-$87.50
-100$

This diagram means, for example, that if the other team leads the
series by two games to one, then you have lost $37.50 so far, and you 
should bet $37.50 on game four (to either get to 2-2 and have $0 if 
your team wins, or to get to 1-3 and be out $75 if your team loses).

Thus you should bet $31.25 on the first game.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/02/2003 at 18:15:34
From: Don
Subject: Thank you (Betting strategy)

Dear Dr. Douglas:

Wonderful! I could hardly believe there is such an answer. Thank you 
so very much. I misunderstood the phrasing. But admittedly the problem 
was too hard for me. Best wishes to you.

Regards,
Don Faulhaber

Date: 05/02/2003 at 19:06:17
From: Doctor Douglas
Subject: Re: Thank you (Betting strategy)

Hi again, Don,

After I wrote out the answer, I was musing to myself about whether I 
like this betting scheme better, or the bet where we simply wager $100 
on who wins the whole series. The outcome of course is exactly the 
same. I think this scheme is nice because it keeps interest in every 
game high.

Of course, if my team gets to 3-0, I'd like to renegotiate the scheme 
- up $87.50 already, I'd offer to wager $100 on the next game, rather 
than the paltry amount of $12.50.

Cheers, 
- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/02/2003 at 20:42:32
From: Don
Subject: Thank you (Betting strategy)

Dear Dr. Douglas:

What is cute about your scheme is the 'dynamics' of the betting 
strategy. At any time in the series, the amount of the bet critically 
depends upon the performance of my home team in previous games to 
date. You are so clever!

Best regards,
Don