OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES.pdf

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 xed 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 nd optimal betting fractions under both forms of bookmaker take. A consequence of

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

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

Introduction

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

investment markets. The British-based on-line nancial market-maker IG Markets oers “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

e ciency and potentially protable betting strategies become more important. To test whether

gambling markets are ine cient 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

conned 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 nance textbook by Luenberger (1998) presents the nancial

theory of investments as eectively 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 ndings. 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 nish before betting on the other.

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

games immediately or before the rst 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 xed-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 dene 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 xed proportion of the bet, multibetting dominates

irrespective of the bettor’s utility function. More specically, 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). 1

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 nd 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 rst 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 prot, 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 i ∈N be the

number of possible outcomes in game i, andO i ={1, 2, . . . , m i }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 i .

For each 1≤k≤n, dene the set,M k , of k -multis by

(1)

M k ={{(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i k , j k )}: j s ∈O i s and i s = i t only if s = t},

and dene the set of all multis,M, by

(2)

M k .

k=1

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

Let γ, γ ′ ∈M. ThenMis equipped with a natural partial order under which γ≺γ ′ if and only if

γ⊂γ ′ . For any γ ={(i 1 , j 1 ), (i 2 , j 2 ), . . . , (i k , j k )}∈M, let

(3)

I γ ={i 1 , . . . , i k }⊂I,

and for i∈I γ dene γ i = j where (i, j)∈γ so that γ i s = j s for 1≤s≤k. Moreover, dene

(4)

p γ =

p i s ,j s =

p i,γ i =

p γ ′ .

s=1

i∈I

γ≺γ ′ ∈M n

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

the bookmaker can be identied with a map α :M→R + , where α γ , for any γ∈M, is the (gross)

payout per dollar bet if the outcome from game i is γ i for all i∈I γ . A gambler in this market

places bets on one or more γ∈Mby wagering an appropriate fraction, f γ , of his wealth on each

such γ. Based on the actual outcome, γ ′ ∈M n , from all the games, the gambler is paid α γ 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∈Iand j∈O i , let ρ i,j be the implied bookmaker probability for the outcome j in

game i, and for any γ∈Mdene

(5)

ρ γ =

ρ i,γ i

ρ γ ′

i∈I

γ≺γ ′ ∈M n

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

on γ is

(6)

α ◦ γ =

ρ γ .

Optimal Strategy Under Fixed Percentage Take

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

actual payout, α γ , for any γ∈M, is

(7)

α γ = (1−ǫ)α ◦ γ ,

OPTIMAL BETTING STRATEGIES FOR SIMULTANEOUS GAMES

where α ◦ γ 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 ofM n .

For each γ∈M, let δ γ be the delta function onMdened by

: 1, if γ ′ = γ,

(8)

δ γ (γ ′ ) =

0, if γ ′ = γ,

and let χ γ :M→R be dened by

(9)

χ γ =

δ γ ′ .

γ≺γ ′ ∈M n

The payout function for a one dollar bet on γ∈Mcan then be written succinctly as α γ χ γ . Note

that for γ∈M n we have χ γ = δ γ .

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

with γ ′ ∈M n and

γ ′ ∈M n f γ ′ = 1 such that

(10)

α γ χ γ =

f γ ′ α γ ′ χ γ ′ .

γ ′ ∈M n

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 n , dene f γ ′ ∈R + by

(11)

f γ ′ =

: α γ /α γ ′ if γ≺γ ′ ,

otherwise.

Then we have

f γ ′ = α γ

α γ ′

= α γ

ρ γ ′

(1−ǫ)

γ ′ ∈M n

γ≺γ ′ ∈M n

α γ

(1−ǫ)

ρ γ ′ =

(1−ǫ) 1

= 1,

α ◦ γ

γ≺γ ′ ∈M n

α γ

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