Also c) How much should you let each of your friends so that you are guarenteed

Question

Also c) How much should you let each of your friends so that you are guarenteed to win $20 regardless of which golfer wins?

3. Arbitrage Betting You and your friends Anna, Bryan and Claire are watching a golf tournament. After 72 holes three players (Leighton, Mcllroy and Norman) have the same total score, so a playoff takes place between them to decide the eventual winner. You and your friends decide to bet on who will win the tournament. (1) Suppose you bet your friend Anna 12:9 against Leiglhton. That is you bet Sz that Leighton (II) Suppose you bet your friend Bryan 9:6 against Mcllroy. That is you bet. Sy that Mcllroy (TII) Suppose you bet your friend Claire 6:4 against Norman. That is, you bet Sz that Norman will lose and if he does Anna owes you $9. If he wins you owe Anna $12. will lose and if he does Bryan owes you S6y. If he wins you owe Bryan $9y. will losc and if he docs Claire owes you S4z. II he wins you owe Claire S6z. (a) Let a represent the money you bet Anna, y represent the money you bet Bryan, and wins. The payout equation corresponding to a Leighton win is shown as an example. Pr =-12x + 6y + 42 z represent the moncy you bet Claire. Using the information in (I), (II) and (III), write three linear equations each describing the payout (money) you will receive if a single player (b) Show that if you bet the same amount of money on all three wagers, i.e. x = y then you are not guaranteed to win any money. $a, z Page 5

Explanation / Answer

a. PL=-12x+6y+4z this equation implies that if Leighton wins then you will lose 12x amount of money to Anna, but since Leighton has won it implies that Mclorry and Norman must have lost so you'd get 6y and 4z dollars from Bryan and Claire repectively.

Similarly if Mclorry wins

PM= 9x-9y+4z (you'll get 9x from Anna(because Leighton lost), lose 9y to Bryan (because Mclorry won)and get 4 from Claire(because Norman lost).

PN = 9x+6y-6z ( 9x from Anna, 6y from Bryan and lose 6z to claire)

b. If you had bet same amount for x,yand z that is equal to a.

Then for PL = -12x+6x+4x= -2x (loss)

For PM=9x-9x+4x= 4x

For PN= 9x+6x-6x= 9x

c. To Make sure that you earn 20 dollars regardless of who wins, then you should decrease the bet with Anna against Leighton from 12:9 to a:9 where a should be anything less than (6y+4y) , then for PL= -ax+6y+4z to be positive,only then you'd get a return = (6y+4z)-ax this difference should be equal to 20 dollars for PM=9x-9y+4z even if x=y=z =a then 4a dollars are guaranteed so a should be equal to 5 dollars and if x,y and z are not equal then (9x+4z) should be greater than 9y such that the difference [(9x+4z)-9y ] is positive and then the valueshould be such that it gives 20 dollars.

For PN=9x+6y-6z if x=y=z= a dollars then you'd 9a dollars

In that case a should be equal to at least 2.23 dollars to get a 20 dollars return. If x,y,z aren't equal then (9x+6y) should be greater than 6z and their difference should be positive such that it gives 20 dollars.

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