BETTING RATES

- CAN THEY BE USEFUL?

Magdalena Hykšová

CTU Prague

2011

Which pair has the greates chance for the win

in the competition StarDance 2010?

e.g., Kříž/Stodůlková: stake 1 EUR win 2,55 EUR or nothing

Similarly in the following cases:

Where should we bet for the football match

Spain – Honduras

at the world championship 2010?

Betting shop 1 (S wins) X (draw) 2 (H wins)

Bet-at-home 1,11 8,00 15,00

Fortuna 1,12 6,70 13,00

Tipsport 1,13 8,10 15,05

Numbers in the table:

how much the bookmaker’s pays for 1 EUR in the case of the sake on the right result

Where should we bet for the football match

Spain – Honduras

at the world championship 2010?

Betting shop 1 (S wins) X (draw) 2 (H wins)

Bet-at-home 1,11 8,00 15,00

Fortuna 1,12 6,70 13,00

Tipsport 1,13 8,10 15,05

Whatever happens, Tipsport pays the most

The highest chance for the win: Spain

We bet for all alternatives to obtain 100 EUR in any case.

How many percent of the stake will we get back?

Betting shop 1 X 2 Return

Bet-at-home 1,11 8,00 15,00

Fortuna 1,12 6,70 13,00

Tipsport 1,13 8,10 15,05

Return – e.g., for Fortuna:

To win 100 EUR in the case of the win of Spain,

we have to bet 100/1,12 on it

In total we pay: 00,13

1100

70,6

1100

12,1

1100Z

We bet for all alternatives to obtain 100 EUR in any case.

How many percent of the stake will we get back?

Betting shop 1 X 2 Return

Bet-at-home 1,11 8,00 15,00 91,53 %

Fortuna 1,12 6,70 13,00 89,36 %

Tipsport 1,13 8,10 15,05 93,04 %

Return – e.g., for Fortuna:

In total we pay: 00,13

1100

70,6

1100

12,1

1100Z

% 36,898936,0

00,13

1

70,6

1

12,1

11

Z

100

betting rate p

... how much the bettor has to bet on the event A to win 1 EUR, if A occurs

Hockey championschip 2010, final CZE – RU

pCZE = 1/4, pRU = 9/10

To win S = 100 CZK, I have to bet

100 1/4 = 25 EUR ... win of the Czech Republic

100 9/10 = 90 EUR ... win of Russia

% 87115

100

betting rate p

... how much the bettor has to bet on the event A to win 1 EUR, if A occurs

Hockey championschip 2010, final CZE – RU

pCZE = 1/4, pRU = 9/10

To win S = 100 CZK, I have to bet

100 1/4 = 25 EUR ... win of the Czech Republic

100 9/10 = 90 EUR ... win of Russia

I will pay 115 EUR, win 100 EUR

return:

Hockey championschip 2010, final CZE – RU

pCZE = 1/4, pRU = 9/10

To win S = 100 CZK, I have to bet

100 1/4 = 25 EUR ... win of the Czech Republic

100 9/10 = 90 EUR ... win of Russia

I will pay 115 EUR, win 100 EUR

return:

pCZE + pRU > 1 the bettor loses

fair bett

% 87115

100

FAIR BETT

the bookmaker specifying p must be in danger of getting into the role of the better

we admit positive and negative betts values

the bettor pays: pCZE S + pRU S, obtains: S

pCZE + pRU > 1 ... bettor chooses S < 0 and wins

pCZE + pRU < 1 ... bettor chooses S > 0 and wins

pCZE + pRU = 1

FAIR BETT

S ... the value of the win if A occurs

(positive or negative)

Bettor’s profit: Z(A) = S – pS = (1 – p) S

Z(A) = – pS

In total we can write: Z = (|A| – p) S

For n events:

Z = (|A1| – p1) S + (|A2| – p2) S + ... + (|An| – pn) S

Bookmaker’s estimation of probability P(A):

e.g., P(A) = 0,5

no danger of changing the roles

it is profitable to propose e.g. p = 0,75

Consider S = 100

win 100 CZK if A occurs ... bett 75 EUR

repetition: bookmaker’s guess about bettor’s profit

ca. 1/2 cases: A occurs Z = 100 – 75 = 25 EUR

ca. 1/2 cases: A does not Z = – 75 EUR

Average profit: URE 25755,0255,0Z

Bookmaker’s estimation of probability P(A):

Mean bettor’s profit:

S > 0 mean bettor’s profit Z < 0

bookmaker benefits

fair bett

Fair bett: we allow S < 0

p ... probability that the bookmaker frankly assigns to the event A

S)p)A(P()pS))(A(P1(S)p1)(A(PZ

BETTING RATES DETERMINATION

bettor’s profit: Z(A) = S – pS = (1 – p) S

Z(A) = – pS

coherent betting rates:

profit Z cannot be always positive, no matter

which event occurs (sure win of the bettor)

coherence requirement basic axioms of probability theory

i.e., to prevent a sure loss, bookmaker’s rates p

have to satisfy axioms of probability theory

BETTING RATES

Warning – who can profit from them?

Application of probability

– determination of rates

(although the bets are not fair)

The meaning of axioms of probability theory

Broadening of probability conception

SUBJECTIVE INTERPRETATION

PROBABILITY  

= measure of personal conviction or belief

in the occurence of some event

Václav Šimerka (1818 – 1887), 1882

Frank Plumpton Ramsey (1903 – 1930), 1931 (1926)

Bruno de Finetti (1906 – 1985), 1931, 1937

Leonard Jimmie Savage (1917 – 1971), 1954

SUBJECTIVE INTERPRETATION

Everyday considerations

• No traffic jam probably occurs on the highway this time.

• Our hockey team has no chance against Russia.

• This drug should be effective in your case.

• The election will be probably won by CSSD.

• If I set out at 16:10, I will perhaps catch the train.

• CNB predicts the 1.8% VAT grow in 2012.