Answer and solution
The bets on the first two games have to be $312.50 each game.
How in the world did I come up with a weird number like that?!?
In explaining this, I will use a picture, since I'm a visual thinker. But hopefully I'll use enough words that you verbal thinkers will follow as well.
The attached picture is a graphical representation of the World Series, with the scores in games of each city along the axes.
Each dot in this graph represents a series standing, e.g., the circled dot represents the situation where city X holds a 2-1 lead. Each of these dots also has associated with it a dollar amount of where the bets so far stand. It is not apparent at this point that that dollar amount is unique (the bets at 0-0, 1-0, and 2-0 might result in a different standing at 2-1 than those at 0-0, 0-1, and 1-1 create) but we will see later that they are unique.
Each line in this graph is really an arrow going toward the right or up, representing the results of a game. Each arrow is associated with a bet, and the dollar amount of the arrow going up or right from a given point is the same (even-money bets).
At this point, we know the dollar standing associated with 9 of these points, $0 at 0-0, $1,000 at the four points in the right most column (x winning the series 4-3, 4-2, 4-1, or 4-0, and ($1,000) at the four points on the top row (We'll look at all dollar amounts from X's perspective.)
Now consider when the series is tied 3-3. If x loses, we go up to ($1,000), while if x wins, we go to the right, to $1,000. The only way this can happen is if the bettors' positions were even at 3-3, and the bet was $1,000. This is the principle used to fill in the whole grid. For any unknown point, if we know the standing that needs to exist at the points above and to the right of that point, then the point in question is midway between those two points, and the bet is half the difference between those two points.
Let's apply that to the point where the score is 3-2. We know that if X wins, he needs to be up by 1,000, and from the previous step, we know that if he loses, he and y need to be at even money. Halfway between $0 and $1,000 is $500,so x must be ahead by $500 at 3-2. Also, ($1,000-$0)/2 = $500 so the wager at 3-2 has to be $500.
One more time. If the score in games is 3-1, the two possible outcomes are X up by $1,000 at 4-1, or X up by $500 at 3-2. So at 3-1, x must be up by $750, and they bet $250 on the subsequent game.
Filling in the rest of the grid is just following this example, which I just did on the lower right, since there is obvious symmetry with Y ahead.
The odd-looking results are that you bet $312.50 on the first two games (0-0 or 1-0), $375 if the series is at 1-1 or 2-1, $500 if the series is at 2-2 or 3-2, $250 if the series stands at 2-0 or 3-1, $125 if the series stands at 3-0, and $1,000 on a game seven.
Or find someone else to escrow your bet, who will do the whole series at once!!
