OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

ANDREW GRANT

∗

, DAVID JOHNSTONE, AND OH KANG KWON

Discipline of Finance H69, The University of Sydney, Sydney NSW 2006, Australia

[a.grant, d.johnstone, o.kwon]@econ.usyd.edu.au

Abstract.

We consider the problem of optimal betting on simultaneous games when the book-

maker accepts bets on the joint outcome of those events, called parlays, accumulators or multibets.

When the bookmaker’s take is a fixed proportion of the amount wagered, multibetting dominates

whatever the bettor’s utility function. When, more commonly, the bookmaker sets multiplicative

payouts, and hence takes a higher percentage on multibets than single bets, the optimal betting

strategy depends on the bettor’s utility function. We consider the case of a log utility (Kelly)

bettor, and find optimal betting fractions under both forms of bookmaker take. A consequence of

these results is that when the bookmaker offers multiplicative payouts, a Kelly bettor’s expected

payout is the same whether the games are simultaneous or sequential.

1. Introduction

Betting markets are growing rapidly and are no longer distinct even superficially from other

investment markets. The British-based on-line financial market-maker IG Markets offers “binary

bets” on whether the London Stock Exchange goes up or down on the day, and brokers 3 million

trades per year at bid and ask prices quoted continuously during the day. Levitt (2004) notes that

turnover of the four major British bookmakers in 2002 exceeded £10 billion. Most betting is on

sports events such as football games but other events such as elections, the Academy Awards, Nobel

Prizes, poker tournaments, and essentially anything with an uncertain outcome, is the subject of

international betting.

As the size and liquidity of betting markets grow, long standing questions about their economic

efficiency and potentially profitable betting strategies become more important. To test whether

gambling markets are inefficient in the sense that they yield positive abnormal returns, it is neces-

sary to design optimal betting portfolios. Most previous discussion of gambling strategies has been

confined to a subculture of mathematicians and professional gamblers, for example Thorp (2000)

Date

: Current version: February 15, 2007.

∗

Corresponding Author.

A. GRANT, D. JOHNSTONE, AND O. KWON

and MacLean, Ziemba and Blazenko (1992). There is, however, an increasing awareness of the

theoretical, practical and linguistic parallels between gambling and other methods of investment

(cf. Levitt (2004), p223-224). The finance textbook by Luenberger (1998) presents the financial

theory of investments as effectively an overriding theory of gambling.

This paper is concerned with the problem of how to bet optimally on two or more activities or

“games” at the same time. We use the term “game” generally to describe the sports event, chess

game, political election, stock market movement or whatever other activity is being bet upon. The

term “game” is therefore not to be taken literally, although such a narrow focus would not alter

the relevance of our findings. We regard “game” as a better term than “event” because an event

can be thought of as either an activity such as a tennis match or as the outcome of that activity,

and may therefore cause confusion.

A problem for gamblers or bettors (terms used interchangeably) is that some games occur con-

temporaneously or at times such that it is not possible to bet on them sequentially. Rather, the

gambler must either bet on one game only or bet on two or more games at the same time. This

occurs for example when two football games overlap in time or when they occur in another time

zone at local times when the gambler cannot wait for one to finish before betting on the other.

Other possibilities are that the gambler may want to secure the fixed betting odds available on both

games immediately or before the first is played, rather than risk these deteriorating beforehand or

between games. Bets must then be made together.

A gambler may bet on any combination of outcomes if the bookmaker is willing to accept such

“multiple” bets. The bettor’s stake is lost if any leg of the multiple bet does not occur. A winning

multiple bet under a fixed-odds system usually sees the bettor receive the product of the individual

games’ contractual (gross) payouts per dollar staked. The bookmaker thus treats the games as

independent. In Britain this type of betting is called an accumulator, in the United States it is

known as a parlay bet, and elsewhere it is simply a multibet, or multi. Given n simultaneous games,

we define a k-multi, for 1 ≤ k ≤ n, to be a multibet on k games.

In this paper, we consider the problem of determining the optimal betting strategy when multi-

betting is available. This is an asset allocation problem, viz. “How can the bettor apportion his

bankroll over single games and their conjunctions such that his expected utility is maximized?”.

We show that if the bookmaker’s “take” is a fixed proportion of the bet, multibetting dominates

irrespective of the bettor’s utility function. More specifically, in the case of n simultaneous games,

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

the optimal strategy consists of betting only on n-multis. A practical example of this type of

multibet is the “daily double” at U.S. racetracks; see Ali (1979), Asch and Quandt (1987), Thaler

and Ziemba (1988).

By comparison, if the bookmaker’s take is multiplicative the optimal betting strategy depends

on the bettor’s utility function. In the case of a log utility (Kelly) bettor, we demonstrate that the

optimal betting strategy requires multibets on all subsets of the n simultaneous games for which the

bettor has positive expected dollar-return. For example, in the case of i = 3 simultaneous games,

each with j = 2 possible outcomes, the bettor must make seven separate bets. These include one

3-multi, three 2-multis, and three 1-multis (bets on single games). We find an analytic solution for

the optimal fraction of wealth for a Kelly bettor to risk on each combined outcome, for any number

of games, each with any number of outcomes. This set of optimal fractions is simply the product of

each of the individual game Kelly fractions, f

i,j

, and complements, b

i,j

= 1 − f

i,j

. Continuing the

above example, the optimal fraction of wealth for the bettor to risk on the 3-multi is the product

of the three optimal Kelly (1956) fractions for individual games. The optimal fraction to be risked

on the first 2-multi (the combination of game 1 and game 2) is the product of the individual game

Kelly fractions for games 1 and 2, multiplied by the complement of the Kelly fraction on game 3.

The pattern continues similarly. Interestingly, the log-optimal betting strategy for n simultaneous

games produces the same profit, and hence utility, as if the bettor was able to bet log-optimally on

all n games sequentially.

2. Notation and Definitions

Suppose there are n ∈ N simultaneous games. For each i ∈ I = {1, 2, . . . , n}, let m

∈ N be the

number of possible outcomes in game i, and O

= {1, 2, . . . , m

} the corresponding set of possible

outcomes. The bettor’s subjective probability of outcome j in game i is denoted by p

i,j

, 1 ≤ j ≤ m

For each 1 ≤ k ≤ n, define the set, M

, of k-multis by

= {{(i

, j

), (i

, j

), . . . , (i

, j

)} : j

∈ O

and i

= i

only if s = t},

(1)

and define the set of all multis, M, by

M =

k=1

(2)

A. GRANT, D. JOHNSTONE, AND O. KWON

Let γ, γ

′

∈ M. Then M is equipped with a natural partial order under which γ ≺ γ

′

if and only if

γ ⊂ γ

′

. For any γ = {(i

, j

), (i

, j

), . . . , (i

, j

)} ∈ M, let

= {i

, . . . , i

} ⊂ I,

(3)

and for i ∈ I

define γ

= j where (i, j) ∈ γ so that γ

= j

for 1 ≤ s ≤ k. Moreover, define

s=1

i∈I

i,γ

γ≺γ

′

∈M

′

(4)

It is assumed that the bookmaker accepts bets on any element of M. The contractual payout by

the bookmaker can be identified with a map α : M → R

, where α

, for any γ ∈ M, is the (gross)

payout per dollar bet if the outcome from game i is γ

for all i ∈ I

. A gambler in this market

places bets on one or more γ ∈ M by wagering an appropriate fraction, f

, of his wealth on each

such γ. Based on the actual outcome, γ

′

∈ M

, from all the games, the gambler is paid α

per

dollar bet on γ if and only if γ ≺ γ

′

. Note that the gambler is not required to place a bet on all

γ ∈ M.

For each i ∈ I and j ∈ O

, let ρ

i,j

be the implied bookmaker probability for the outcome j in

game i, and for any γ ∈ M define

i∈I

i,γ

γ≺γ

′

∈M

′

(5)

in analogy with (4). Then in the absence of a bookmaker take, it would be the case that the payout

on γ is

◦

(6)

3. Optimal Strategy Under Fixed Percentage Take

In this section, suppose that the bookmaker takes a fixed fraction, ǫ > 0, of each bet so that the

actual payout, α

, for any γ ∈ M, is

= (1 − ǫ)α

◦

(7)

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

where α

◦

is the payout in the absence of bookmaker take as in (6). We show below that, irrespective

of the utility function of the gambler, the optimal strategy under (7) consists of betting only on

n-multis, viz. only on elements of M

For each γ ∈ M, let δ

be the delta function on M defined by

(γ

′

) =











if γ

′

= γ,

if γ

′

6= γ,

(8)

and let χ

: M → R be defined by

γ≺γ

′

∈M

′

(9)

The payout function for a one dollar bet on γ ∈ M can then be written succinctly as α

. Note

that for γ ∈ M

we have χ

= δ

Lemma

1: Let the bookmaker payouts be given by (7). Then for any γ ∈ M, there exist f

′

∈ R

with γ

′

∈ M

and

′

∈M

′

= 1 such that

′

∈M

′

(10)

That is, the payouts from a bet on any γ ∈ M can be replicated by a suitable combination of bets

on n-multis.

Proof

. For any γ

′

∈ M

, define f

′

∈ R

′











/α

′

if γ ≺ γ

′

otherwise.

(11)

Then we have

′

∈M

′

= α

γ≺γ

′

∈M

′

= α

γ≺γ

′

∈M

′

(1 − ǫ)

γ≺γ

′

∈M

′

(1 − ǫ)

◦

= 1,

A. GRANT, D. JOHNSTONE, AND O. KWON

where the second last equality follows from (5) and (6), and the final equality follows from the

definition of α

. Moreover, (10) holds by the definition of f

′

in (11), and so f

′

have the stated

properties.

Q.E.D

Theorem

1: Let the bookmaker payouts be given by (7). Then for any k ∈ N

, and γ

∈ M and

∈ R

, with 1 ≤ i ≤ k and

k
i=1

≤ 1, there exist f

′

∈ R

with γ

′

∈ M

and

′

∈M

′

≤ 1

such that

i=1

′

∈M

′

(12)

That is, the payouts from any combination of bets on elements of M can be replicated by a suitable

combination of bets on n-multis.

Proof

. For each 1 ≤ i ≤ k, there exist f

′

∈ R

such that

′

∈M

′

= 1 and

′

∈M

′

by the previous lemma. Let f

′

k
i=1

′

∈ R

. Then

′

∈M

′

∈M

i=1

′

i=1

′

∈M

′

i=1

· 1 ≤ 1,

and we have

i=1

′

∈M

′

∈M

i=1

′

∈M

′

Hence, the f

′

have the stated properties.

Q.E.D

Corollary

1: If the bookmaker takes a fixed fraction of each bet, then a strategy consisting of

bets on n-multis dominates any other betting strategy. Moreover, this does not depend on the utility

function of the gambler.

Proof

. Theorem 1 shows that any betting strategy can be replicated by a strategy consisting of

bets only on n-multis.

Q.E.D

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

If the gambler is a log utility maximizer, that is, he maximizes ¯

U = E[ln(w)] of terminal wealth w,

then the fractions, f

′

with γ

′

∈ M

, for an optimal betting strategy can be determined according

to the rules in Kelly (1956).

Example

1: In this example, we demonstrate how a bet on elements of M

(that is, on single

games) can be replicated using bets on 2-multis in the case of two games, each with two outcomes.

Firstly a $1 bet on {(1, 1)} ∈ M

has payout

(1,1)

1 − ǫ
ρ

(1,1)

(13)

where for notational convenience we have set α

(i,j)

= α

{(i,j)}

and ρ

(i,j)

= ρ

{(i,j)}

. By Lemma 1, this

can be replicated using a combination (pair) of bets on 2-multis, {(1, 1), (2, 1)} and {(1, 1), (2, 2)},

with fractions

(1,1)

{(1,1),(2,1)}

= ρ

(2,1)

and

(1,1)

{(1,1),(2,2)}

= ρ

(2,2)

(14)

Note that f

(1,1)

{(1,1),(2,1)}

and f

(1,1)

{(1,1),(2,2)}

sum to 1 (so the total bet remains $1) and the payouts

received under the two approaches are identical, irrespective of the game outcomes. Similarly, a

bet of $1 on {(2, 1)} ∈ M

with payout

(2,1)

1 − ǫ
ρ

(2,1)

(15)

can be replicated using

(2,1)

{(1,1),(2,1)}

= ρ

(1,1)

and

(2,1)

{(1,2),(2,1)}

= ρ

(1,2)

(16)

Consider now a bet on {(1, 1)} and {(2, 1)} with fractions f

and f

respectively of $1, where

+ f

≤ 1. By Theorem 1, this can be replicated using a combination of bets on 2-multis,

{(1, 1), (2, 1)}, {(1, 1), (2, 2)}, and {(1, 2), (2, 1)}, with fractions

{(1,1),(2,1)}

= f

(1,1)

{(1,1),(2,1)}

+ f

(1,1)

{(1,1),(2,1)}

= f

(2,1)

+ f

(2,2)

(17)

{(1,1),(2,2)}

= f

(2,1)

{(1,1),(2,1)}

= f

(2,1)

(18)

{(1,2),(2,1)}

= f

(2,1)

{(1,2),(2,1)}

= f

(1,2)

(19)

A. GRANT, D. JOHNSTONE, AND O. KWON

The corresponding payouts on the 2-multis are

{(1,1),(2,1)}

1 − ǫ

(1,1)

(2,1)

(20)

{(1,1),(2,2)}

1 − ǫ

(1,1)

(2,2)

(21)

{(1,2),(2,1)}

1 − ǫ

(1,2)

(2,1)

(22)

and it is easily verified that, irrespective of the outcomes on each game, the payouts using the two

approaches coincide.

4. Log Optimal Strategy Under Multiplicative Take

In practice, bookmakers, such as Multibet,

do not take a fixed fraction of each bet, but instead

provide payouts on γ ∈ M that are products of payouts on individual games. For any γ ∈ M this

corresponds to assigning

= (1 − ǫ)

i∈I

◦

i,γ

(23)

where α

◦

i,γ

is the payout on {(i, γ

)} ∈ M

in the absence of a bookmaker take and |I

| is the size

of I

. In this case, Corollary 1 no longer applies. In fact, it is not even possible to replicate the

payoffs on elements of M

using a strategy involving n-multis.

To see this, let γ = {(i, j)} ∈ M

, where i ∈ I and j ∈ O

, and consider the random variable

= (1 − ǫ)α

◦

(24)

which represents the payoff on betting on the event that the outcome from game i is j. In order to

find a strategy involving only the n-multis that dominates (or replicates) this payoff, we must find

′

∈ R

, with

′

∈M

′

≤ 1, such that

(1 − ǫ)α

◦

≤ (1 − ǫ)

′

∈M

′

◦

′

(25)

In particular, for any γ

′

∈ M

with γ ≺ γ

′

, we must have

(1 − ǫ)α

◦

≤ (1 − ǫ)

′

◦

′

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

which implies

′

≥

(1 − ǫ)

n−1

◦

′

(1 − ǫ)

n−1

′

But then we must have

′

∈M

′

≥

′

∈M

: γ≺γ

′

≥

(1 − ǫ)

n−1

′

∈M

: γ≺γ

′

(1 − ǫ)

n−1

· 1 > 1,

which shows that it is not possible to find such f

′

. We now determine the optimal betting strategy

under (23) for the special case of a gambler with the log utility function, known after Kelly (1956)

as a “Kelly bettor”.

For any betting strategy {f

: γ ∈ M}, where f

is the fraction of wealth bet on γ, and for any

γ ∈ M let

= b +

γ≻γ

′

∈M

′

(26)

where

b = 1 −

γ∈M

(27)

Note then that for the log utility gambler the function to be maximized is

U (f ) =

γ∈M

ln Γ

(28)

Theorem

2: For each i ∈ I, let {f

∗

i,j

: j ∈ O

}, be the optimal Kelly fractions for a single game

bet, and let b

∗

= 1 −

j∈O

∗

i,j

. Then for a gambler with the log utility function, the optimal betting

strategy, {f

∗

: γ ∈ M}, is

∗

i∈I

∗

i,γ

i /

∈I

∗

(29)

for γ ∈ M, and the corresponding fraction of wealth not bet is b

∗

i∈I

∗

Proof

. Let O

= {j ∈ O

: f

∗

i,j

> 0}, O

−

= {j ∈ O

: f

∗

i,j

= 0} = O

and M

= {γ ∈ M : γ

∈

for all i ∈ I

}. Then since f

∗

6= 0 if and only if γ ∈ M

, it follows from the discussion in Boyd

A. GRANT, D. JOHNSTONE, AND O. KWON

and Vandenberghe (2004), subsection 4.2.3, that it suffices to show

∂ ¯

U (f

∗

)

∂f

= 0, if γ ∈ M

(30)

∂ ¯

U (f

∗

)

∂f

≤ 0, if γ /

∈ M

(31)

Firstly, note that for any γ

∈ M we have

∂ ¯

U (f

∗

)

∂f

≺γ∈M

−

γ∈M

(32)

Computing the second term of (32), which is independent of γ

, we obtain

γ∈M

∗

′

≺γ

′

∗

′

γ∈M

i∈I

i,γ

∗

+ α

i,γ

∗

i,γ

i=1





j∈O

+
i

i,j

∗

+ α

i,j

∗

i,j

j∈O

−
i

i,j

∗





But for any i ∈ I and j ∈ O

, we have from Kelly (1956) that

i,j

∗

+ α

i,j

∗

i,j

− f

∗

i,j

∗

(33)

and so

γ∈M

i=1





j∈O

+
i

i,j

− f

∗

i,j

∗

j∈O

−
i

i,j

∗





i=1

∗





j∈O

i,j

−

j∈O

+
i

∗

i,j





i=1

∗





1 −

j∈O

+
i

∗

i,j





= 1,

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

since b

∗

= 1 −

j∈O

+
i

∗

i,j

. Computing the first term of (32) for an arbitrary γ

∈ M, we obtain

≺γ∈M

∗

′

≺γ∈M

′

∗

′

≺γ∈M

∗

′

≺γ∈M

′

∗

′

≺γ∈M

i∈I

i,γ

∗

+ α

i,γ

∗

i,γ

i∈I

γ0

i,γ

0 i

i,γ

∗

+ α

i,γ

0 i

∗

i,γ

i /

∈I

γ0





j∈O

+
i

i,j

∗

+ α

i,j

∗

i,j

j∈O

−
i

i,j

∗





i∈I

γ0

i,γ

0 i

i,γ

∗

+ α

i,γ

0 i

∗

i,γ

i /

∈I

γ0

i∈I

γ0

: γ

0 i

∈O

+
i

i,γ

0 i

i,γ

∗

+ α

i,γ

0 i

∗

i,γ

i∈I

γ0

: γ

0 i

∈O

−
i

i,γ

∗

i∈I

γ0

: γ

0 i

∈O

−
i

i,γ

∗

where the final equality follows from (33). If γ

∈ M

, it follows from definitions that {i ∈

: γ

∈ O

−

} = ∅ and so

≺γ∈M

= 1.

Hence, ∂ ¯

U (f

∗

)/∂f

= 0 for γ

∈ M

which proves (30). Now, if γ

∈ M

then {i ∈ I

: γ

∈

−

} 6= ∅, and since α

i,j

≤ b

∗

for all i ∈ I and j ∈ O

−

by Kelly (1956), we have

≺γ∈M

i∈I

γ0

: γ

∈O

−
i

i,γ

∗

≤ 1.

Hence, ∂ ¯

U (f

∗

)/∂f

≤ 0 for γ

∈ M

which proves (31). Finally, it is easily verified that b

∗

γ∈M

∗

= 1, and this completes the proof.

Q.E.D

It follows from the theorem that the optimal strategy in this case does not consist entirely of bets

on n-multis. Moreover, in contrast to the single game case in which the optimal betting fractions

may be non-zero for a negative expectation outcome, the optimal betting fractions are non-zero

only for elements of γ ∈ M

which consists entirely of bets with positive expected return.

A. GRANT, D. JOHNSTONE, AND O. KWON

As in the proof of Theorem 2, let f

∗

i,j

be the optimal single game Kelly fractions, b

∗

= 1 −

j∈O

∗

i,j

and b

∗

i∈I

∗

. We now compute the optimal expected log utility, ¯

U (f

∗

), assuming

initial wealth w

= 1.

U (f

∗

) =

γ∈M

ln Γ

γ∈M





∗

′

≺γ

′

∗

′





γ∈M





i∈I

: γ

∈O

+
i

∗

+ α

i,γ

∗

i,γ

) ·

i∈I

: γ

∈O

−
i

∗





(34)

γ∈M





i∈I

: γ

∈O

+
i

i,γ

i∈I

: γ

∈O

−
i

∗





γ∈M





i∈I

i,γ

i∈I

: γ

∈O

−
i

∗

i,γ





γ∈M

ln p

γ∈M

ln α

γ∈M

ln β

where

i∈I

: γ

∈O

−
i

∗

i,γ

The next theorem shows that for any outcome, γ ∈ M

, on the n games, the terminal wealth for

the optimal M-strategy coincides with the terminal wealth where the games are played in sequence

and the optimal Kelly fractions are wagered on each single game.

Theorem

3: Suppose the games, i ∈ I, are played one after the other and the optimal single

game Kelly fractions are wagered on each game. Then for any set of outcomes, γ ∈ M

, the

corresponding terminal wealth coincides with the terminal wealth on the log optimal M-strategy.

Proof

. It is well known that for any sequence of outcomes, γ ∈ M

, the terminal wealth of a

log utility (Kelly) bettor is independent of the order in which the games are played. Hence, the

terminal wealth, ˜

, corresponding to γ ∈ M

and initial wealth w

= 1, is

i∈I

: γ

∈O

+
i

∗

+ α

i,γ

∗

i,γ

) ·

i∈I

: γ

∈O

−
i

∗

But from (34), this is precisely the terminal wealth, w

, for the optimal M-strategy.

Q.E.D

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

Interestingly, therefore, the terminal wealth for the log-optimal bet on a set of games does not

depend on whether the games are played simultaneously or in sequence. In other words, the only

rationale for a Kelly gambler betting sequentially, rather than simultaneously, in a set of sequential

games, is that there may be a favorable change in the fixed odds for one or more of those games

before it is played.

5. An Empirical Comparison

A common betting strategy involves betting only on elements of M

, which corresponds to

betting on single games. In this section, we consider the benefits of multibetting relative to betting

on individual games. For analytic tractability, we restrict the analysis to the special case of two

games, each with two outcomes and with symmetric payouts. The analysis, however, can be

extended to the general case of an arbitrary number of games and outcomes by numerical means.

In the case of two games with symmetric payouts, the optimal M

-strategy can be easily obtained.

The quoted payout data on December 11, 2005, for two NFL games in Table 1, was obtained

from multibet, an online bookmaker that accepts multibets and offers multiplicative payouts as

per sports betting market convention. We denote by f

∗

the optimal M-strategy, and by f

†

the

optimal M

-strategy.

Table 1.

Bookmaker Payouts for NFL Games

Home Team

Away Team

(i,1)

(i,2)

Cincinnati Bengals

Cleveland Browns $1.10

$8.00

2 San Diego Chargers

Miami Dolphins

$1.10

$8.00

Expected (log) utility for the optimal M-strategy is given in Figure 1 for a range of subjective

probabilities. The highest value is obtained when both subjective probabilities approach 1. This is

the point where the majority of the bettor’s wealth is invested in the 2-multi, {(1, 1), (2, 1)}, which

represents the bet on the joint outcome of Cincinnati and San Diego both winning.

The difference in expected utility under the optimal M-strategy and the optimal M

-strategy,

U (f

∗

) − ¯

U (f

†

), is shown in Figure 2. Clearly, the optimal multibetting strategy performs at least

as well as the M

strategy over the entire range of probabilities, p

(1,1)

and p

(2,1)

. In particular, as

the bettor’s subjective probabilities both approach 1, the optimal multibetting strategy increases

its superiority over the optimal M

strategy.

A. GRANT, D. JOHNSTONE, AND O. KWON

Figure 1.

Expected Log Utility for the Optimal M-Strategy.

0.9

0.92

0.94

0.96

0.98

0.9

0.92

0.94

0.96

0.98

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

(1,1)

(2,1)

¯ U

∗

)

As an example of the calculation of a particular point from Figure 2, suppose that the bettor’s

subjective probabilities of the home teams, Cincinnati and San Diego, winning are p

(1,1)

= 0.97

and p

(2,1)

= 0.95, respectively. Then the optimal betting fractions under the optimal M and M

strategies are as given in Table 2. Note that a large proportion of the bettor’s wealth is transferred

Table 2. Betting Fractions and Expected Log Utility for the Optimal

M and M

Strategies

{(1, 1)} {(2, 1)} {(1, 1), (2, 1)}

∗

0.3685

0.1485

0.3015

0.04157

†

0.6055

0.3327

–

0.03880

to the 2-multi from 1-multis, {(1, 1)} and {(2, 1)}, when the bettor shifts from an optimal M

strategy to the optimal M-strategy.

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

Figure 2.

Difference in Expected Log Utilities for the Optimal M and M

Strategies

0.9

0.92

0.94

0.96

0.98

0.9

0.92

0.94

0.96

0.98

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(1,1)

(2,1)

¯ U

∗

)

−

¯ U

†

)

6. Concluding Remarks

We consider the problem of a gambler facing simultaneous betting opportunities against a book-

maker offering fixed payouts. A theoretical framework for multibetting is established under the

assumption that the bookmaker accepts bets not only on single games, but also on the joint out-

come of multiple games. We determine the optimal betting strategy for n simultaneous games

under the two most common types of bookmaker take, proportional and multiplicative.

It is shown that the optimal strategy under the proportional bookmaker take consists solely of

n-multis and that this is independent of the bettor’s utility function. Moreover, it is possible to

replicate any betting strategy involving lower-dimensional multis with a strategy that employs only

n-multis. In the case of a log utility gambler, optimal betting fractions can be found using the

results from Kelly (1956).

A. GRANT, D. JOHNSTONE, AND O. KWON

When the bookmaker offers multiplicative payouts, which is generally the case in the fixed-

odds market, the optimal betting strategy depends on the gambler’s utility function. It is shown

that, under log utility, the optimal multibetting fractions can be expressed simply in terms of the

corresponding optimal single game fractions. In particular, the optimal fractions are positive only

for those multis with positive dollar-expectation. It is also shown that, for any combination of

outcomes, the profit from the log optimal (Kelly) bet does not depend on whether the games are

played simultaneously or in sequence.

Endnotes

We assume fixed odds betting, whereas daily double bets at U.S. racetracks are generally

parimutuel, meaning that the market betting odds are determined after the race, based on the

total amount in the betting pool.

http://www.multibet.com

Levitt (2004, p.223) comments that the price adjustments tend to be “small and infrequent” in

betting markets, relative to the frequent price movements in financial markets. Similarly, Kuypers

(2000) finds that the U.K. soccer fixed odds rarely change once they have been set.

Insley, Mok and Swartz (2004) have provided such an example.

References

Ali, M. M. (1979), ‘Some Evidence on the Efficiency of a Speculative Market’, Econometrica 47, 387–392.

Asch, P. and Quandt, R. E. (1987), ‘Efficiency and Profitability in Exotic Bets’, Economica 54, 289–298.

Boyd, S. and Vandenberghe, L. (2004), Convex Optimization, Cambridge University Press, Cambridge, U.K.

Insley, R., Mok, L. and Swartz, T. (2004), ‘Issues Related to Sports Gambling’, Australian and New Zealand Journal

of Statistics 46

, 219–232.

Kelly, J. L. (1956), ‘A New Interpretation of Information Rate’, Bell Systems Technical Journal 35, 917–926.

Kuypers, T. (2000), ‘Information and Efficiency: An Empirical Study of a Fixed Odds Betting Market ’, Applied

Economics 32

, 1353–1363.

Levitt, S. D. (2004), ‘Why Are Gambling Markets Organised So Differently From Financial Markets?’, The Economic

Journal 114

, 223–246.

Luenberger, D. (1998), Investment Science, Oxford University Press, New York.

ENDNOTES

MacLean, L. C., Ziemba, W. T. and Blazenko, G. (1992), ‘Growth Versus Security in Dynamic Investment Analysis’,

Management Science 38

, 1562–1585.

Thaler, R. H. and Ziemba, W. T. (1988), ‘Anomalies: Parimutuel Betting Markets: Racetracks and Lotteries’, Journal

of Economic Perspectives 2

, 161–174.

Thorp, E. O. (2000), The Kelly Criterion in Blackjack, Sports Betting and the Stock Market, in O. Vancura, J. A.

Cornelius and W. R. Eadington, eds, ‘Finding The Edge: Mathematical Analysis of Casino Games’, Institute

For the Study of Gambling and Commercial Gaming, Reno, NV, pp. 163–213.

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