1. Nov 19, 2013

### mz19

1. The problem statement, all variables and given/known data

1. Sports arbitrage: The following lines are available at different bookmakers on next week’s game for the Steelers

versus the Ravens:

Bookmaker 1: 1:1 odds that the Oakland beats Tennessee (if you bet $1, you win$2 for an Oakland win)

Bookmaker 2: 2:1 odds that the Tennessee beats the Oakland (if you bet $1, you receive$3 for a

Tennessee win)

You have $1000 to spend and bets must be made at each bookmaker in multiples of$100. If X is the amount that

you place with Bookmaker 1 and Y is the amount that you place at Bookmaker 2:

a. Draw the feasible region for combinations of bets that will guarantee a WINNING (not breakeven)

amount and label its corners. Use solid or dotted lines appropriately.

b. Choose a bet combination that will guarantee a positive payout after the game is played. You’ll need

to go to the interior or far edge of the region to find one that the bookmakers will accept.

2. Relevant equations

Just the information given and general math knowledge

3. The attempt at a solution

So I know that to set this up, x + y ≤ $1000 I'm unsure of where to incorporate the$2 and $3 winnings into an equation If it has to be in multiples of$100 to equal the $1000 than can I just use x+y≤100 because there are 100 parts? As in he place 100 bets divided between x and y. If it has to be winning and not break even then x($2) + y($3) >$1000. It cannot be equal

2. Nov 19, 2013

### Mentallic

Actually, since the bets can only be made in $100 chunks, then you basically have 10 "chunks" to work with right? So if we denote x and y to be how many bets or "chunks" we give to bookmaker 1 and 2 respectively then we have $$x+y\leq 10$$ Now, to help us figure out what to do next with the bets, lets look at what happens when we make certain bets. (x,y) will be how many$100 bets we put into x, and how many we put into y.

If we bet (10,0) then this means we've put all our money on Oakland (O) winning. If O wins, we win 2*10 = 20 (20 chunks of bets or \$2000). If T wins, we get 3*0 = 0

(9,1) -> If O wins, we get 2*9 = 18, if T wins, we get 3*1 = 3

(8,2) -> O wins 2*8 = 16, T wins 3*2 = 6

(7,3) -> O wins 2*7 = 14, T wins 3*3 = 9

(6,4) -> O wins 2*6 = 12, T wins 3*4 = 12

Notice at this point that no matter who wins, we get >10 in return from both the wins from x or y! So what are the 2 equations? What have we been doing to x and y separately to find the winnings?

Also consider that if we do something like (6,3) then when O wins, we get 2*6 = 12 and T wins gives us 3*3 = 9, but since we only put 9 bets in, we have 1 bet left over and hence our winnings will be 13 and 10 respectively.

Last edited: Nov 19, 2013