Evaluating betting odds and free coupons using desirability
Nawapon Nakharutai, Camila C. S. Caiado, Matthias C. M. Troffaes
aa r X i v : . [ q -f i n . M F ] J a n EVALUATING BETTING ODDS AND FREE COUPONS USINGDESIRABILITY
NAWAPON NAKHARUTAI, CAMILA C. S. CAIADO, AND MATTHIAS C. M. TROFFAES
Abstract.
In the UK betting market, bookmakers often offer a free couponto new customers. These free coupons allow the customer to place extra bets,at lower risk, in combination with the usual betting odds. We are interestedin whether a customer can exploit these free coupons in order to make a suregain, and if so, how the customer can achieve this. To answer this question,we evaluate the odds and free coupons as a set of desirable gambles for thebookmaker.We show that we can use the Choquet integral to check whether this setof desirable gambles incurs sure loss for the bookmaker, and hence, results ina sure gain for the customer. In the latter case, we also show how a customercan determine the combination of bets that make the best possible gain, basedon complementary slackness.As an illustration, we look at some actual betting odds in the market andfind that, without free coupons, the set of desirable gambles derived fromthose odds avoids sure loss. However, with free coupons, we identify somecombinations of bets that customers could place in order to make a guaranteedgain. Introduction
Consider the football betting market in the UK where a bookmaker typicallyoffers fractional betting odds for possible outcomes. For example, in a match be-tween Manchester United and Liverpool, the bookmaker offers odds in the form a/b for Manchester United winning, c/d for a draw and e/f for Liverpool winning.Suppose a customer accepts the odds a/b by placing a stake of b pounds on aManchester United win, which he pays to the bookmaker in advance of the match.After the match, if Manchester United wins, the bookmaker will pay him a + b pounds. So, if Manchester United wins, then the customer’s total return will be a pounds; otherwise the customer will lose b pounds.To predict the outcome of a match, the bookmaker may encounter difficultiessuch as lack of data (e.g. team A has never played with team B during last fiveyears), missing data, limited football expert opinion, or even contradicting infor-mation from different football experts. Various authors [14, 15, 13, 10] have arguedthat these issues can be handled by using sets of desirable gambles . A gamble rep-resents a reward (i.e. money in our case) that depends on an uncertain outcome(i.e. the match result). The bookmaker can model his belief about this outcomeby stating a collection of gambles that he is willing to offer. Such set is calleda set of desirable gambles. Through duality, stating a set of desirable gambles ismathematically equivalent to stating a set of probability distributions. Key words and phrases. betting; coupon; Choquet integration; complementary slackness.
If there are no combinations of desirable gambles that result in a guaranteed loss,then we say that a set of desirable gambles avoids sure loss [14, 15]. Thus, if thebookmaker’s set of desirable gambles avoids sure loss, then there is no combinationof bets from which customers can make a guaranteed gain. On the other hand, ifthe set does not avoid sure loss, then there is a combination of bets that customerscan exploit to incur a sure gain.In addition to avoiding sure loss, the bookmakers also want to entice new cus-tomers. There are several techniques that bookmakers can use to persuade cus-tomers to bet with their companies. Some bookmakers may offer greater bettingodds than others since greater odds means a greater payoff to the customers. An-other technique is to offer a “free coupon”, which is a stake that customers canspend on betting. The free coupon can also be viewed as part of a desirable gam-ble.However, bookmakers may worry that customers will find a combination of dif-ferent odds and free coupons that they can bet on and make a guaranteed profit.Therefore, from the bookmaker’s perspective, they would like to check whether setsof desirable gambles derived from different odds and free coupons avoid sure lossor not. Conversely, in theory, a customer may be interested in the case where thebookmaker’s set does not avoid sure loss, because then the customer can make aguaranteed profit. In that case, a customer may want to find the combination ofbets which results in the best possible sure gain.There are several studies on exploiting betting odds and free bets in order to findstrategies that make a profit. For example, Walley [13, Appendix I] and Quaeghe-beur et al. [7] study an application of sets of desirable gambles on sports; Millineret al. [5], Schervish et al. [9], Vlastakis et al. [12] exploit betting odds directly,whilst Emiliano [2] takes free bets into account. Emiliano considers the case of onlytwo possible outcomes, and allows cooperation between customers. In this paper,we look at any finite number of possible outcomes, but we only consider a singlecustomer. We evaluate betting odds and free coupons and check whether a set ofdesirable gambles derived from odds and free coupons avoids sure loss (or not) viathe natural extension. If the set does not avoid sure loss, then we show exactly howa customer can incur a sure gain.In general, one can check avoiding sure loss by solving a linear programmingproblem [13, p. 151]. In our previous work [6], we provided efficient algorithms forsolving these linear programming problems. For our specific problem, we show thatwe can calculate the natural extension through the Choquet integral, or throughsolving a linear programming problem where the optimal value is equal to thenatural extension. In the case of not avoiding sure loss, we know that we can find astrategy that the customer can bet on to make a guaranteed gain. We show that thisstrategy can be identified using the Choquet integral and complementary slacknessconditions. Our method for finding this strategy is generally applicable not just tothis betting problem, but to arbitrary problems involving upper probability massfunctions. Specifically, by using the Choquet integral and exploiting complementaryslackness conditions, we can find optimal solutions of the corresponding pair of duallinear programming programs without directly solving them.The paper is organised as follows. Section 2 briefly reviews the main conceptsbehind desirability, avoiding sure loss and natural extension. We also discuss theChoquet integral which can be used to calculate the natural extension. In section 3,
VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 3 we introduce fractional fixed odds and explain how betting odds work. As bettingodds can be viewed as a set of desirable gambles, we revisit a simple known algo-rithm to check whether such set avoids sure loss or not. In section 4, we discuss freecoupons from the perspective of desirability. We show how we can check whetherthe problem with free coupons avoids sure loss or not, by means of the naturalextension. We demonstrate how we can use the Choquet integral to calculate thisnatural extension. Next, we exploit complementary slackness to find a combinationof bets which makes the best possible guaranteed gain. To illustrate our results, insection 5, we consider some actual betting odds and free coupons in the market,and provide an example where a customer can make a sure gain with a free coupon.Section 6 concludes this paper.2.
Avoiding sure loss and natural extension
In this section, we will briefly discuss desirability, avoiding sure loss and naturalextension. We will also explain the Choquet integral which can be used to calculatethe natural extension in the case considered in this paper. The material in thissection will be useful later when we view betting odds and free coupons as a set ofdesirable gambles and when we want to check whether this set avoids sure loss ornot.2.1.
Avoiding sure loss.
Let Ω be a finite set of uncertain outcomes. A gamble is a bounded real-valued function on Ω. Let L (Ω) denote the set of all gambles onΩ. Let D be a finite set of gambles that a subject deems acceptable; we call D thesubject’s set of desirable gambles . Rationality conditions for desirability have beenproposed as follows [10, p. 29]: Axiom 1 (Rationality axioms for desirability) . For every f and g in L (Ω) andevery non-negative α ∈ R , we have that: (D1) If f ≤ and f = 0 , then f is not desirable. (D2) If f ≥ , then f is desirable. (D3) If f is desirable, then so is αf . (D4) If f and g are desirable, then so is f + g . The first two axioms are trivial as the subject should accept any gamble that hecannot lose from, but he should not accept any gamble that he cannot win from.Axiom (D3) follows the linearity of the utility scale and axiom (D4) shows that acombination of desirable gambles should also be desirable.We do not assume that any set D , specified by the subject, satisfies these axioms.However, we can use these axioms to examine the rationality of D . Indeed, therationality axioms essentially state that a non-negative combination of desirablegambles should not produce a sure loss [10, p. 30]. In that case, we say that D avoids sure loss. Definition 1. [10, p. 32] A set
D ⊆ L (Ω) is said to avoid sure loss if for all n ∈ N ,all λ , . . . , λ n ≥ , and all f , . . . , f n ∈ D , (1) max ω ∈ Ω n X i =1 λ i f i ( ω ) ! ≥ . Note that the rationality axioms for desirability are stronger than the conditionof avoiding sure loss [10, p. 32].
NAWAPON NAKHARUTAI, CAMILA C. S. CAIADO, AND MATTHIAS C. M. TROFFAES
We can also model uncertainty via acceptable buying (or selling) prices for gam-bles. A lower prevision P is a real-valued function defined on some subset of L (Ω).We denote the domain of P by dom P . Given a gamble f ∈ dom P , we interpret P ( f ) as a subject’s supremum buying price for f , i.e. f − α is deemed desirable forall α < P ( f ) [10, p. 40]. Definition 2. [10, p. 42] A lower prevision P is said to avoid sure loss if for all n ∈ N , all λ , . . . , λ n ≥ , and all f , . . . , f n ∈ dom P , (2) max ω ∈ Ω n X i =1 λ i [ f i ( ω ) − P ( f i )] ! ≥ . Any lower prevision P induces a conjugate upper prevision P on − dom P := {− f : f ∈ dom P } , defined by P ( f ) := − P ( − f ) for all f ∈ − dom P . P ( f )represents a subject’s infimum selling price for f [10, p. 41].Next, let A denote a subset of Ω, also called an event . Its associated indicator function I A is given by(3) ∀ ω ∈ Ω : I A ( ω ) := ( ω ∈ A . Further in the paper, we will also extensively use upper probability mass func-tions . An upper probability mass function p is a mapping from Ω to [0 , ∀ ω ∈ Ω : P p ( − I { ω } ) := − p ( ω ) , where dom P p = S ω ∈ Ω {− I { ω } } . We can check whether P p avoids sure loss bytheorem 1. Theorem 1. [10, p. 124] P p avoids sure loss if and only if P ω ∈ Ω p ( ω ) ≥ .Proof. See [10, p. 124, Prop. 7.2] with lower probability mass function p = 0. (cid:3) We can interpret an upper probability mass function as providing an upperbound on the probability of each { ω } , for all ω ∈ Ω [10, p. 123].2.2.
Natural extension.
The natural extension of a set of desirable gambles D isdefined as the smallest set of gambles which includes all finite non-negative combi-nations of gambles in D and all non-negative gambles [10, § Definition 3. [10, p. 32] The natural extension of a set
D ⊆ L (Ω) is: (5) E D := ( g + n X i =1 λ i g i : g ≥ , n ∈ N , g , . . . , g n ∈ D , λ , . . . , λ n ≥ ) . From this natural extension, we can derive a supremum buying price for anygamble f . Definition 4. [10, p. 46] For any set
D ⊆ L (Ω) and f ∈ L (Ω) , we define: E D ( f ) := sup { α ∈ R : f − α ∈ E D } (6) = sup ( α ∈ R : f − α ≥ n X i =1 λ i f i , n ∈ N , f i ∈ D , λ i ≥ ) . (7) VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 5
Note that E D is finite, and hence, is a lower prevision, if and only if D avoidssure loss [10, p. 68].We denote the conjugate of E D by E D which is defined by(8) E D ( f ) := − E D ( − f ) = inf ( β ∈ R : β − f ≥ n X i =1 λ i f i , n ∈ N , f i ∈ D , λ i ≥ ) . for all f in L (Ω) [13, p. 124]. E D is simply denoted by E when there is no confusion.Given a lower prevision P , we can derive a set of desirable gambles correspondingto P as follows [10, p. 42]:(9) D P := { g − µ : g ∈ dom P and µ < P ( g ) } . Combining definition 4 and eq. (9) together, we can define the natural extension of P : Definition 5. [10, p. 47] Let P be a lower prevision. The natural extension of P is defined for all f ∈ L (Ω) by: (10) E P ( f ) := E D P ( f )= sup ( α ∈ R : f − α ≥ n X i =1 λ i ( f i − P ( f i )) , n ∈ N , f i ∈ dom P , λ i ≥ ) . Similarly, E P is finite if and only if P avoids sure loss [10, p. 68].In the next section, we briefly explain the use of the Choquet integral to calculatethe natural extension for the type of lower previsions considered in this paper; see[11, 10] for more detail.2.3. Upper probability mass functions and Choquet integration.
Let E p be the natural extension of P p that avoids sure loss. Then E p is 2-monotone andcan be computed via the Choquet integral [10, p. 125]. In this section, based onthe results from [10, Sec. 7.1], we give a closed form expression for this integral.For simplicity, we denote the natural extension E p ( I A ) of an indicator I A as E p ( A ). We can use the following theorem to calculate E p ( A ). Theorem 2. [10, p. 125] Let P p avoid sure loss. Then for all A ⊆ Ω , (11) E p ( A ) = max { , − U ( A c ) } and E p ( A ) = min { U ( A ) , } , where U ( A ) := P ω ∈ A p ( ω ) . Proof.
See [10, p. 125] with lower probability mass function p = 0. (cid:3) Theorem 3.
Let f be decomposed in terms of its level sets A i , i = 0 , , . . . , n : (12) f = n X i =0 λ i I A i where λ ∈ R , λ , . . . , λ n > and Ω = A ) A ) · · · ) A n = ∅ . Then (13) E p ( f ) = n X i =0 λ i E p ( A i ) . Proof.
The right hand side is the Choquet integral [10, p. 379, Eq. (C.8)] and thenatural extension E p ( f ) is equal to the Choquet integral [10, p. 125, Prop. 7.3(ii)](with lower probability mass function p = 0). (cid:3) NAWAPON NAKHARUTAI, CAMILA C. S. CAIADO, AND MATTHIAS C. M. TROFFAES
Note that theorem 3 also holds for the upper natural extension.
Corollary 1.
Let f be a gamble decomposed as in eq. (12) . Then (14) E p ( f ) = n X i =0 λ i E p ( A i ) . Proof.
See appendix A. (cid:3)
The Choquet integral will be useful when we want to calculate the natural ex-tension later in section 4.2.4.
Avoiding sure loss with one extra gamble.
Let D = { g , . . . , g n } be a setof desirable gambles that avoids sure loss and let f be another desirable gamble.We want to check whether D ∪ { f } still avoids sure loss or not. This idea will beused when we want to check avoiding sure loss with a free coupon in section 4.By the condition of avoiding sure loss in definition 1, D ∪ { f } avoids sure loss ifand only if for all λ ≥ n ∈ N , g i ∈ D and λ , . . . , λ n ≥ ω ∈ Ω n X i =1 λ i g i ( ω ) + λ f ( ω ) ! ≥ . We can simplify eq. (15) as follows.
Lemma 1.
Let Ω be a finite set, D = { g , . . . , g n } be a set of desirable gamblesthat avoids sure loss and f be another desirable gamble. Then, D ∪ { f } avoids sureloss if and only if for all n ∈ N , g i ∈ D and λ , . . . , λ n ≥ , (16) max ω ∈ Ω n X i =1 λ i g i ( ω ) + f ( ω ) ! ≥ . Proof. If λ = 0 in eq. (15), then eq. (15) is trivially satisfied because D avoids sureloss. Otherwise λ >
0, and for all i , λ i ≥
0, so λ i /λ ≥
0. Therefore eq. (15) isequivalent to(17) max ω ∈ Ω n X i =1 (cid:18) λ i λ (cid:19) g i ( ω ) + f ( ω ) ! ≥ . Therefore,
D ∪ { f } avoids sure loss if and only if eq. (16) holds. (cid:3) Next, we give a method not only for checking avoiding sure loss of
D ∪ { f } , butalso for bounding the worst case loss, which will be useful later in section 4. Theorem 4.
Let f ∈ L (Ω) and let D = { g , . . . , g n } be a set of desirable gamblesthat avoids sure loss. Then, D ∪ { f } avoids sure loss if and only if E D ( f ) ≥ . If D ∪ { f } does not avoid sure loss, then there exist λ ≥ , . . . , λ n ≥ such that f + P ni =1 λ i g i , which is a combination of desirable gambles, results in a loss at least | E D ( f ) | .Proof. See appendix B. (cid:3)
Note that by definition 5, theorem 4 can also be applied to E P . VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 7 Betting scheme
In this section, we explain how fractional betting odds work and look at twoscenarios: (i) a customer bets against a bookmaker and (ii) a customer bets againstmultiple bookmakers. In both cases, we view betting odds as a set of desirablegambles and check whether such a set avoids sure loss or not.3.1.
Betting with one bookmaker.
In the UK, a bookmaker usually offers fixedfractional odds on possible outcomes of an event that customers are interestedin. For example, in the European Football Championship 2016, customers areinterested in the winner of the championship. Suppose that a bookmaker sets oddson France, say 9 /
2, and one customer accepts this odds. For every stake £ £ £ £
2. The bookmaker often writes a/ a .Given fractional odds a/b , a customer can simply calculate his return as follows.For every amount b that the customer bets, he will either get nothing (in case thebet is lost), or gain a plus the return of his stake (in case the bet is won). Asthe bookmaker accepts this transaction, the total payoff can be seen as a desirablegamble, say g , to the bookmaker:(18) g ( ω ) = ( − a if ω = xb otherwise . Note that − g is a desirable gamble to the customer, should the customer decide toaccept the bookmaker’s odds.Let Ω = { ω , . . . , ω n } be a finite set of outcomes. Suppose that for each i , thebookmaker sets betting odds a i /b i on ω i . By eq. (18), these odds can be viewed asa set of desirable gambles D = { g , . . . , g n } , where(19) g i ( ω ) := ( − a i if ω = ω i b i otherwise . Given odds a i /b i on ω i , suppose that we modify the denominator in this odds tobe b j . To do so, we can multiply a i /b i by b j /b j to be(20) a i b j /b i b j = (cid:18) a i b j b i (cid:19) /b j . Are new odds still desirable? By the rationality axioms for desirability, the modifiedodds are still desirable.
Lemma 2.
Let a/b be odds on an outcome ˜ ω that are desirable. Then, for all α > , the odds αa/αb on ˜ ω are also desirable.Proof. Consider the desirable gamble corresponding to the odds a/b :(21) g ( ω ) := ( − a if ω = ˜ ωb otherwise . By rationality axiom (D3), for any α >
0, the gamble αg is also desirable. Hence,the corresponding odds αa/αb are also desirable. (cid:3) NAWAPON NAKHARUTAI, CAMILA C. S. CAIADO, AND MATTHIAS C. M. TROFFAES
Lemma 2 will be very useful when we want to modify odds to have the samedenominator.Suppose that the bookmaker specifies betting odds for all possible outcomes inΩ. Before announcing these odds, the bookmaker may want to check whether thereis a combination of bets from which the customer can make a sure gain, or in otherwords, whether he avoids sure loss [13, Appendix 1, I4, p. 635]:
Theorem 5.
Let
Ω = { ω , . . . , ω n } . Suppose a i /b i are betting odds on ω i . Foreach i ∈ { , . . . , n } , let (22) g i ( ω ) := ( − a i if ω = ω i b i otherwisebe the gamble corresponding to the odds a i /b i . Then D := { g , . . . , g n } avoids sureloss if and only if (23) n X i =1 b i a i + b i ≥ . Proof.
Theorem 5 follows from theorem 6 (proved further) for m = 1. (Note thattheorem 5 is not used in the proof of theorem 6.) (cid:3) Note that, in practice, P ni =1 b i a i + b i is normally strictly greater than 1, and(24) 100 × n X i =1 b i a i + b i − ! is called the over-round margin [2, 12].Let’s see an example of theorem 5. Example 1.
Suppose that a bookmaker provides betting odds / for W, / forD, and / for L. As (25) 43 + 4 + 513 + 5 + 516 + 5 = 1 . ≥ , by theorem 5, the bookmaker avoids sure loss. Therefore, a customer cannot exploitthese odds in order to make a sure gain. Note that the condition for avoiding sure loss of D in theorem 5 is exactly thesame as the condition for avoiding sure loss of P p in theorem 1. This condition isalso equivalent to Proposition 4 in Cortis [1].Next, we show that those odds can be modelled through an upper probabilitymass function: Lemma 3.
Let
Ω = { ω , . . . , ω n } , let ω i ∈ Ω and let g be the corresponding gambleto the odds on ω i defined as in eq. (19) , that is, (26) g i ( ω ) := ( − a i if ω = ω i b i otherwise , where a i and b i are non-negative. If p is a probability mass function, that is, if P ω ∈ Ω p ( ω ) = 1 and p ( ω ) ≥ for all ω ∈ Ω , then (27) X ω ∈ Ω g i ( ω ) p ( ω ) ≥ ⇐⇒ b i a i + b i ≥ p ( ω i ) . VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 9
Proof.
Suppose that P ω ∈ Ω p ( ω ) = 1 and for all i, p ( ω i ) ≥
0, then X ω ∈ Ω g i ( ω ) p ( ω ) ≥ ⇐⇒ − a i p ( ω i ) + b i X ω = ω i p ( ω ) ≥ ⇐⇒ − a i p ( ω i ) + b i (1 − p ( ω i )) ≥ ⇐⇒ b i a i + b i ≥ p ( ω i ) . (30) (cid:3) In order to avoid sure loss, the odds a i /b i on ω i must satisfy eq. (30) [13, § ∀ i ∈ { , . . . , n } : p ( ω i ) := b i a i + b i . Betting with multiple bookmakers.
In the market, there are many book-makers. We are interested in whether a customer can exploit odds from differentbookmakers in order to make a sure gain. To do so, we model betting odds fromdifferent bookmakers as a set of desirable gambles, and we check avoiding sure lossof this set. We recover the known result that it is optimal to pick maximal odds oneach outcome [12]. As greater odds correspond to a higher payoff to a customer, asensible strategy for him is to pick the greatest odds on each outcome.
Theorem 6.
Let
Ω = { ω , . . . , ω n } . Suppose there are m different bookmakers. Foreach k ∈ { , . . . , m } , let a ik /b ik be the betting odds on ω i provided by bookmaker k .For each i ∈ { , . . . , n } and k ∈ { , . . . , m } , let (32) g ik ( ω ) := ( − a ik if ω = ω i b ik otherwise . be the desirable gamble corresponding to the odds a ik /b ik . Let a ∗ i /b ∗ i be the maximalbetting odds on outcome ω i , that is, (33) a ∗ i /b ∗ i := m max k =1 { a ik /b ik } . Then the set of desirable gambles D = { g ik : i ∈ { , . . . , n } , k ∈ { , . . . , m }} avoidssure loss if and only if (34) n X i =1 b ∗ i a ∗ i + b ∗ i ≥ . Proof.
See appendix C. (cid:3)
Theorem 6 tells us that to check avoiding sure loss of several bookmakers, weonly need to consider the maximal odds on each outcome. Let’s see an example.
Example 2.
Suppose that in the market there are three bookmakers providing dif-ferent odds for outcomes W, D, and L as in table 1.
Outcomes Betting companies Maximum oddsRiver Mountain ForestW / /
20 3 / / D / / / / L / / / Table 1.
Table of odds provided by three bookmakers
Let D be the set of desirable gambles corresponding to all of these odds. Notethat the maximal betting odds are / for W, / for D and / for L. As (35) 2017 + 20 + 514 + 5 + 310 + 3 = 1 . ≥ , by theorem 6, we conclude that D avoids sure loss. Therefore, a customer cannotexploit these odds to make a sure gain. Consider a customer who is interested in odds provided by the three bookmakersas in table 1. A sensible strategy to him is to pick the greatest odds on each out-come. However, this means that the customer will never choose any odds providedby Forest, because all of Forest’s odds are less than the odds provided by otherbookmakers. Therefore, to encourage customers to bet with them, Forest may offerfree coupons to the customer under certain conditions. In the next section, we willlook at these free coupons in more detail.4.
Free coupons for betting
A free coupon is a free stake that is given by a bookmaker to a customer whofirst bets with him. The free coupon can be spent on some betting odds thatthe customer wants to bet. In fact, the free coupon is not truly free, since thecustomer firstly has to bet on some odds before he claims the free coupon. Moreover,the bookmakers usually set some required conditions, for instance, a limit on theamount of free coupons that customers can claim, or a restriction of choices thatcustomers can spend their free coupons.We were wondering whether customers can exploit those given odds and freecoupons in order to find a strategy of betting that incurs a sure gain. If there is apossible way to do that, then we will find an algorithm that gives such a strategy.For simplicity in this study, we set up standard requirements for claiming freecoupons from the bookmakers as follows:(1) Once the customer has placed his first bet, the bookmaker will give him afree coupon whose value is equal to the value of the bet that he placed.(2) The bookmaker sets the maximum value of the free coupon.(3) The free coupon only applies to the customer’s first bet with the bookmaker.(4) The customer must spend his free coupon with the same bookmaker onother outcomes.(5) The customer must spend his free coupon on only a single outcome.Here is an example of claiming free coupons.
Example 3.
Suppose that Forest has the following offer: a free coupon will be givento a customer who first bets with Forest, and the value of the coupon is equal to thevalue of the first bet that the customer placed.
VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 11
From table 1, if James, who is a customer, has never bet with Forest and hedecides to place £ on the odds / of the outcome D, then he will play £ toForest and he will claim a free coupon valued £ . James can use the free couponto bet on other outcomes with Forest. Once James receives a free coupon, he can spend his free coupons as in the nextexample.
Example 4.
Continuing from the previous example, James has his free couponvalued £ from Forest. Since James must spend his free coupon valued £ on onlya single outcome, by lemma 2, we modify odds / by multiplying them by / . Nowall odds have the same denominator which is .Outcomes W D Lodds ( · ) / / / Table 2.
Table of modified odds
If James spends his free coupon to bet on L and the true outcome is L, then Forestwill lose £ ; otherwise Forest will lose nothing. On the other hand, if Jamesspends the coupon to bet on W and the true outcome is W, then Forest will lose£ · ; otherwise Forest will lose nothing. A total payoff to Forest is summarised intable 3. Betting a free coupon on OutcomesW D LL − W − · Table 3.
Table of total payoff
Suppose that the customer first bets on an outcome ω i with corresponding odds a i /b i . The payoff to the bookmaker is represented as a gamble g ω i in the table 4.Because this is his first bet, the customer receives a free coupon valued b i , and hewill spend this free coupon to bet on a single outcome. Suppose that he bets on ω j with corresponding odds a j /b j . As the denominators are not necessarily equal, wemultiply odds a j /b j by b i b i . The modified odds are ( a j · b i b j ) /b i . Note that as the freecoupon must be spent on other outcomes, ω j cannot coincide with ω i .If the true outcome is ω j , then the bookmaker will lose a j · b i b j . Otherwise thebookmaker will gain nothing. This payoff to the bookmaker is viewed as a gamble˜ g ω j in the table 4. As g ω i and ˜ g ω j are desirable to the bookmaker, by rationalityaxiom (D4), g ω i + ˜ g ω j is also desirable.Outcomes ω i ω j others g ω i − a i b i b i ˜ g ω j − a j · b i b j g ω i + ˜ g ω j − a i ( b j − a j ) b i b j b i Table 4.
Table of the first-free desirable gamble to the bookmaker
We denote g ω i ω j := g ω i + ˜ g ω j and call it the first-free desirable gamble to thebookmaker. Note that − g ω i ω j is desirable to the customer. The customer can bet on other odds, but he will not get any free coupon from his additional bets. Thisis because the bookmaker gives him the free coupon only once.Also note that in the actual market, there is usually more than one bookmakeroffering a free coupon. Therefore, the customer can first bet with different book-makers in order to obtain several free coupons. These can be viewed as a first-freedesirable gamble combining from several first-free desirable gambles. In this study,we only consider the case that customer first bets and claims a free coupon from asingle bookmaker. In this case, we face a combinatorial problem over all first-freedesirable gambles.We would like to check whether D∪{ g ω i ω j } avoids sure loss or not. By theorem 4,if D avoids sure loss, then D ∪ { g ω i ω j } avoids sure loss if and only if E ( g ω i ω j ) ≥ D ∪ { g ω i ω j } does not avoid sure loss, by theorem 4, the bookmakerwill lose at least | E ( g ω i ω j ) | which is the customer’s highest sure gain. Therefore,the customer can combine g ω i ω j with a non-negative combination of g i to obtain asure gain | E ( g ω i ω j ) | .Let f be any first-free desirable gamble to the bookmaker. Before using theresults in Section 2.3 to calculate the natural extension of f , we have to checkwhether D avoids sure loss. If P p does not avoid sure loss, then without a freecoupon, there is a non-negative combination of gambles that the customer canexploit to make a sure gain. On the other hand, if P p avoids sure loss, then wecan write f in terms of its level sets and use corollary 1 to calculate the naturalextension of f . Example 5.
Let Forest provide betting odds on W, D, and L as in table 1. Byeq. (31) , we have (36) p ( W ) = 47 p ( D ) = 518 p ( L ) = 521 . Since p ( W ) + p ( D ) + p ( L ) ≥ , P p avoids sure loss by theorem 1.Continuing from example 4, suppose that James first bets on D and spends hisfree coupon to bet on L. Then, the first-free desirable gamble g DL to Forest is asfollows: Outcomes W D Lg D −
13 5 g L − g DL − − Table 5.
Table of desirable gambles to Forest
We decompose g DL in terms of its level sets as (37) g DL = − I A + 2 I A + 16 I A where A = { W, D, L } , A = { W, L } and A = { W } . By theorem 2, we have E p ( A ) = min { p ( W ) + p ( D ) + p ( L ) , } = 1(38) E p ( A ) = min { p ( W ) + p ( L ) , } = 1721(39) E p ( A ) = min { p ( W ) , } = 47 . (40) VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 13
Substitute E p ( A i ) , i ∈ { , , } into eq. (37) . By corollary 1, we have (41) E p ( g DL ) = − E p ( A ) + 2 E p ( A ) + 16 E p ( A ) = − . As E p ( g DL ) = − < , by theorem 4, Forest does not avoid sure loss. Therefore,with the free coupon, James can make a sure gain. How should James bet? Remember that Ω = { ω , . . . , ω n } and that g i is thecorresponding gamble to the odds a i /b i on ω i :(42) g i ( ω ) = ( − a i if ω = ω i b i otherwise . Note that we can calculate E p ( f ), or E D Pp ( f ), by definition 5, for any gamble f by solving the following linear program:(P) min α (Pa) subject to ( ∀ ω ∈ Ω : α − P ni =1 g i ( ω ) λ i ≥ f ( ω ) ∀ i = 1 , . . . , n : λ i ≥ , (Pb)where the optimal α gives E p ( f ). If the optimal α is strictly negative, then theoptimal λ , . . . , λ n give a combination of bets for a customer to make a sure gain.The dual of (P) is(D) max X ω ∈ Ω f ( ω ) p ( ω )(Da) subject to ∀ g i : P ω ∈ Ω g i ( ω ) p ( ω ) ≥ ∀ ω : p ( ω ) ≥ P ω ∈ Ω p ( ω ) = 1 . (Db1)After applying lemma 3, the constraints in eq. (Db1) become:subject to ( ∀ ω : 0 ≤ p ( ω ) ≤ p ( ω ) P ω ∈ Ω p ( ω ) = 1 . (Db2)We see that the objective function eq. (Da) is E p ( f ), the expectation of f withrespect to the probability mass function p . As the optimal value of (D) is E p ( f ),if we can find a p that satisfies the dual constraints eq. (Db2) and E p ( f ) = E p ( f ),then we have found an optimal solution of (D).We now first construct a p , by assigning as much mass as possible to the smallestlevel sets. Then, in theorem 7, we prove that this p satisfies eq. (Db2) and E p ( f ) = E p ( f ). Algorithm 1
Construct an optimal solution p of (D) Input:
A gamble f , a set of outcomes Ω. Output:
An optimal solution p of (D).(1) Rewrite f as(43) f = m X i =0 λ i A i where Ω = A ) A ) · · · ) A m ) ∅ are the level sets of f and λ ∈ R , λ , . . . , λ m > (2) Order ω , ω , . . . , ω n such that(44) ∀ i ≤ j : A ω i ⊆ A ω j , where A ω is the smallest level set to which ω belongs, that is(45) A ω = m \ i =0 ω ∈ A i A i . So, we start with those ω in A m , then those in A m − \ A m , then thosein A m − \ A m − , and so on.(3) Let k be the smallest index such that(46) k X j =1 p ( ω j ) ≥ . There is always such k because P p avoids sure loss. Define p as follows: p ( ω i ) := p ( ω i ) if i < k − P i − j =1 p ( ω j ) if i = k i > k. (47)We then show that p in eq. (47) satisfies eq. (Db2) and E p ( f ) = E p ( f ). Theorem 7.
The probability mass function p defined by eq. (47) satisfies eq. (Db2) and E p ( f ) = E p ( f ) .Proof. Let Ω = { ω , . . . , ω n } be ordered as in eq. (44), and let k be the smallestindex such that P kj =1 p ( ω j ) ≥
1. By eq. (47), P ni =1 p ( ω i ) = 1 and(48) p ( ω k ) = 1 − k − X j =1 p ( ω j ) ≤ k X j =1 p ( ω j ) − k − X j =1 p ( ω j ) = p ( ω k ) , so for all i ∈ { , . . . , n } , 0 ≤ p ( ω i ) ≤ p ( ω i ). Therefore, p satisfies eq. (Db2). Next,we will show that for all level sets A i ,(49) min ( X ω ∈ A i p ( ω ) , ) = E p ( A i ) . Remember that A ω k is the smallest level set that contains ω k . By eq. (47), for all A i ( A ω k , we know that p ( ω ) = p ( ω ) for all ω ∈ A i , and so(50) min ( X ω ∈ A i p ( ω ) , ) = X ω ∈ A i p ( ω ) = X ω ∈ A i p ( ω ) . For all A i ⊇ A ω k , we know that P ω ∈ A i p ( ω ) = 1 and P ω ∈ A i p ( ω ) ≥
1, so(51) min ( X ω ∈ A i p ( ω ) , ) = 1 = X ω ∈ A i p ( ω ) . VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 15
Hence, eq. (49) holds. Therefore, E p ( f ) = m X i =0 λ i E ( A i ) (by eq. (14))(52) = m X i =0 λ i min ( X ω ∈ A i p ( ω ) , ) (by eq. (11))(53) = m X i =0 λ i E p ( A i ) (by eq. (49))(54) = E p ( f )(55) (cid:3) To sum up, we can use eq. (47) to construct an optimal solution p of (D).We will use complementary slackness to find an optimal solution of the dual of(D) [16, p. 329]. Note that, as (D) has an optimal solution and the dual problem isbounded above, then by the strong duality theorem [8, p. 71], an optimal solutionof (P) exists and achieves the same optimal value. In addition, a pair of solutionsto (P) and (D) is optimal if, and only if, they satisfy the complementary slacknesscondition [3, p. 62]. Specifically, in our case, the condition holds for any non-negative variable and its corresponding dual constraint [4, p. 184, ll. 3–5]. More,precisely, let p ( ω ), . . . , p ( ω n ) be any feasible solution of (D), and let α , λ , . . . , λ n be any feasible solution of (P). Then, by complementary slackness, these solutionsare optimal if, and only if, for all j ∈ { , . . . , n } , we have that(56) α − n X i =1 g i ( ω j ) λ i − f ( ω j ) ! p ( ω j ) = 0 and ( p ( ω j ) − p ( ω j )) λ j = 0 . This is equivalent to(1) if p ( ω j ) >
0, then α − P ni =1 g i ( ω j ) λ i = f ( ω j ), and(2) if p ( ω j ) < p ( ω j ), then λ j = 0.So, if we have an optimal solution p ( ω ), . . . , p ( ω n ) of (D) and the optimal value α , then we can use these equations as a system of equalities in λ , . . . , λ n . Notethat some solutions of this system may not satisfy feasibility, i.e. they may violate λ i ≥
0. However, all solutions of this system that satisfy λ i ≥ k was definedas the smallest index such that P kj =1 p ( ω j ) ≥
1. According to eq. (47), for all j ∈ { , . . . , k − } we have that p ( ω j ) >
0, so we have the following equalities: forall j ∈ { , . . . , k − } ,(57) α − n X i =1 g i ( ω j ) λ i = f ( ω j ) . For all j ∈ { k + 1 , . . . , n } we have that p ( ω j ) = 0 < p ( ω j ), so λ j = 0 for all j ∈ { k + 1 , . . . , n } . For j = k , if p ( ω k ) < p ( ω k ), then we can also set λ k = 0.Otherwise, we know that p ( ω k ) = p ( ω k ) > j ∈ { , . . . , k − } . Concluding, let k ′ be the largest index j for which p ( ω j ) = p ( ω j ). Then as the optimal solution of (P) exists, it can be foundby solving the following system: ∀ j ∈ { , . . . , k ′ } : α − k ′ X i =1 g i ( ω j ) λ i = f ( ω j )(58) ∀ j ∈ { k ′ + 1 , . . . , n } : λ j = 0(59)So, effectively, all we are left with is a system of k ′ variables in k ′ constraints.Note that we can modify the odds to have the same denominator (all b i areequal), so it will be much easier to solve the new system.Finally, note that in the first-free coupon scenario, to make a sure gain, thecustomer has to bet on every outcome. This implies that the only coefficients λ i whose value can be zero are those corresponding to the gambles in the first-freegamble chosen by the customer. Hence, in that specific case, k ′ ≥ n − Example 6.
Continuing from example 5, the corresponding linear programs to E ( g DL ) are as follows: (P1) min α (P1a) subject to α + 3 λ W − λ D − λ L ≥ α − λ W + 13 λ D − λ L ≥ − α − λ W − λ D + 16 λ L ≥ − and λ W , λ D , λ L ≥ , α free , (P1c) (D1) max 5 p ( W ) − p ( D ) − p ( L )(D1a) subject to ≤ p ( W ) ≤ / ≤ p ( D ) ≤ / ≤ p ( L ) ≤ / p ( W ) + p ( D ) + p ( L ) = 1 . (D1b) By eq. (44) , we see that (60) A W ⊆ A L ⊆ A D , so an optimal solution of (D1) is as follows: (61) p ( W ) = 47 , p ( L ) = 521 , p ( D ) = 1 − (cid:18)
47 + 521 (cid:19) = 421 . As p ( W ) = p ( W ) and p ( L ) = p ( L ) , whilst p ( D ) < p ( D ) , by the complementaryslackness, the optimal solution of (P1) must have λ D = 0 and solves the followingsystem: α + 3 λ W − λ L = 5(P1b1) α − λ W + 16 λ L = − , (P1b2) where the value of α is − . We solve this system and get an optimal solution: λ W = and λ L = .A strategy for James to make a guaranteed gain is as follows. He first bets £ on D and claims a free coupon valued £ to bet on L. Next, he additionally bets£ on W and £ on D. He will make a sure gain of £ from Forest. VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 17
Country Odds Country Odds Country OddsFrance 10/3 Austria 45 Czech Republic 135Germany 23/5 Poland 50 Slovakia 150Spain 5 Switzerland 66 Rep of Ireland 170England 9 Russia 85 Iceland 180Belgium 57/5 Turkey 94 Romania 275Italy 91/5 Wales 100 N Ireland 400Portugal 20 Ukraine 100 Hungary 566Croatia 27 Sweden 104 Albania 531
Table 6.
Table of maximum betting odds for the European FootballChampionship 2016 Actual football betting odds
In this section, we will look at some actual odds in the market, and we will checkwhether and how a customer can exploit those odds and free coupons in order tomake a sure gain.Consider table 9 which is in appendix D. We list betting odds provided by 27bookmakers on the winner of the European Football Championship 2016. Fromtable 9, the maximum betting odds on each outcome are listed in table 6. For all i ∈{ , . . . , } , let a ∗ i /b ∗ i be the maximal betting odds in table 6. Since P i =1 b ∗ i a ∗ i + b ∗ i =1 . ≥
1, by theorem 6, the set of desirable gambles corresponding to the oddsin table 9 avoids sure loss. Therefore, there is no combination of bets which resultsin a sure gain.Suppose that James is interested in betting with one of them, say Bet2. As hehas never bet with Bet2 before, Bet2 will give him a free coupon on his first betwith them. With free coupons, we will check whether and how James can bet tomake a guaranteed gain. Let D be a set of desirable gambles corresponding to theodds and let g be any first-free desirable gamble to the company Bet2. We want tocheck whether D ∪ { g } avoids sure loss or not. As there are 24 possible outcomes,the total number of different first-free desirable gambles with Bet2 is 24 ×
23 = 552.Suppose that James first bets on France and then spends his free coupon onSpain. So, the the first-free desirable gamble g F G isOutcomes France Spain others g F − g S − g F S − − Table 7.
James’ first-free gamble where F and S denote France and Spain respectively. Again, we calculate E ( g F S )by the Choquet integral. We decompose g F S in terms of its level sets as(62) g F S = − I A + I A + 4 I A where A = Ω, A = Ω \ { S } and A = Ω \ { F, S } . By theorem 2, we have(63) E ( A ) = 1 E ( A ) = 0 . E ( A ) = 0 . . By corollary 1, we substitute E ( A i ), i ∈ { , , } to eq. (62) and obtain(64) E ( g F S ) = − E ( A ) + E ( A ) + 4 E ( A ) = − . . Therefore,
D ∪ { g F S } does not avoid sure loss.Among all possible first-free gambles, we find that there are three further gam-bles whose E is less than zero, namely E ( g F G ) = − . E ( g GF ) = − . E ( g GS ) = − . G denotes Germany. So, by theorem 4, D ∪ { g } does notavoid sure loss when g ∈ { g F S , g
F G , g GF , g GS } ; otherwise D ∪ { g } avoids sure loss.Therefore, if(1) James first bets on France and then spends his free coupon to bet on eitherSpain or Germany, or(2) James first bets on Germany and then spends his free coupon to bet oneither France or Spain,then there is a combination of bets for him to bet in order to make a sure gain fromBet2.Consider the case where James first bets £ p ( ω i ) in table 8) can be found through algorithm 1. Then, we can find the optimalsolution of the corresponding problem (P) by using the optimal solution of (D) withthe complementary slackness condition. The optimal solution of (P) is presentedin a column λ i in table 8. Therefore, if James additionally bets as in column λ i ,then he will make a sure gain of £ .
095 from Bet2.6.
Conclusion
In this paper, we studied whether and how a customer can exploit given bettingodds and free coupons in order to make a sure gain. Specifically, we viewed theseodds and free coupons as a set of desirable gambles and checked whether such a setavoids sure loss or not via the natural extension. We showed that the set avoidssure loss if, and only if, the natural extension of the first-free gamble correspondingto the free coupon is non-negative. If the set does not avoid sure loss, then acombination of bets can be derived from the optimal solution of the correspondinglinear programming problem.We showed that for this specific problem, we can easily find the natural extensionthrough the Choquet integral. In the case that the set does not avoid sure loss, wepresented how to use the Choquet integral and the complementary slackness con-dition to directly obtain the desired combination of bets, without actually solvinglinear programming problems, but instead just solving a linear system of equalities.This technique can be applied to arbitrary problems involving upper probabilitymass functions.To illustrate the results, we looked at some actual betting odds on the winningof the European Football Championship 2016 in the market, and checked avoidingsure loss. We found that any sets of desirable gambles derived from those oddsavoid sure loss. Having said that, with a free coupon, we identified sets of desirablegambles that no longer avoid sure loss. So, interestingly, in this case, when a free
VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 19
Order ω i Countries Odds p ( ω i ) Optimal solutions p ( ω i ) λ i
15 15
12 England 9
110 110 .
53 Belgium 10
111 111 511
117 117 517
119 119 519
126 126 526
141 141 541
151 151 551
141 141 541
10 Russia 66
167 167 567
11 Turkey 80
181 181 581
12 Wales 80
181 181 581
13 Ukraine 66
167 167 567
14 Sweden 80
181 181 581
15 Czech Republic 100
16 Slovakia 100
17 Rep of Ireland 150
18 Iceland 150
19 Romania 100
20 N Ireland 250
21 Albania 250
22 Hungary 250
23 France 3
14 14 14
24 Spain 5
16 586579 Table 8.
A summary of odds provided by Bet2, the upper probabilitymass function p ( ω i ), and optimal solution of (D) and (P) coupon is added, there was a combination of bets from which the customer couldhave made a sure gain. Acknowledgements
We would like to acknowledge support for this project from Development andPromotion of Science and Technology Talents Project (Royal Government of Thai-land scholarship). We also thank the reviewers for their constructive comments.
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VALUATING BETTING ODDS AND FREE COUPONS USING DESIRABILITY 21
Appendix A. Proof of corollary 1
Proof.
Since A = Ω, we can write f as(65) f = n X i =1 λ i I A i + λ where λ ∈ R , λ , . . . , λ n > A ) · · · ) A n ) ∅ . Then − f = − n X i =1 λ i (1 − I A c i ) − λ = − n X i =1 λ i − λ + n X i =1 λ i I A c i . (66)Therefore, E p ( f ) = − E p ( − f )(67) = − − n X i =1 λ i − λ + n X i =1 λ i E p ( A c i ) ! (68) = λ + n X i =1 λ i (1 − E p ( A c i )(69) = λ + n X i =1 λ i E p ( A i ) , (70)where eq. (68) holds by constant additivity and comonotone additivity [10, p. 382,Prop. C.5(v)&(vii)]. (cid:3) Appendix B. Proof of theorem 4
Proof.
For the first part, suppose that f ∈ L (Ω) and D = { g i : i ∈ { , . . . , n }} is aset of desirable gambles that avoids sure loss. We find that E D ( f ) = inf ( α ∈ R : α − f ≥ n X i =1 λ i g i , λ i ≥ ) = min ( max ω ∈ Ω f ( ω ) + n X i =1 λ i g i ( ω ) ! : λ i ≥ ) , (71)where the inf is actually a min because D is finite. So, by lemma 1,(72) E D ( f ) ≥ ⇐⇒ ∀ λ i ≥ , max ω ∈ Ω n X i =1 λ i g i ( ω ) + f ( ω ) ! ≥ . For the second part, if
D ∪ { f } does not avoid sure loss, then E D ( f ) <
0. So, byeq. (71), there exists an ω ∗ in Ω and some λ i ≥ E D ( f ) = f ( ω ∗ ) + n X i =1 λ i g i ( ω ∗ ) ≥ f ( ω ) + n X i =1 λ i g i ( ω ) , ∀ ω ∈ Ω . Hence there is a sure loss of at least | E D ( f ) | . (cid:3) Appendix C. Proof of theorem 6
Proof.
Note that for each i and k , we have(74) a ik b ik ≤ a ∗ i b ∗ i ⇐⇒ b ∗ i a ∗ i + b ∗ i ≤ b ik a ik + b ik . So,(75) b ∗ i a ∗ i + b ∗ i = min k (cid:26) b ik a ik + b ik (cid:27) . (= ⇒ ) Suppose the set of desirable gambles D avoids sure loss. We will show thateq. (34) holds. As D avoids sure loss, the following system of linear inequalities: ∀ i : p ( ω i ) ≥ n X i =1 p ( ω i ) = 1(77) ∀ i, k : n X i =1 g ik ( ω i ) p ( ω i ) ≥ , (78)has a solution [13, p. 175, ll. 10–13], say p = ( p ( ω ) , . . . , p ( ω n )). By lemma 3, foreach i and k ,(79) b ik a ik + b ik ≥ p ( ω i ) . Then, by eq. (75) for each i ,(80) b ∗ i a ∗ i + b ∗ i ≥ p ( ω i ) . Therefore,(81) n X i =1 b ∗ i a ∗ i + b ∗ i ≥ n X i =1 p ( ω i ) = 1 . ( ⇐ =) Suppose P ni =1 b ∗ i a ∗ i + b ∗ i ≥ S = n X i =1 b ∗ i a ∗ i + b ∗ i and p ( ω i ) = b ∗ i S ( a ∗ i + b ∗ i ) . If we show that p is a feasible solution of eqs. (76), (77) and (78), then D avoids sureloss. Note that by eq. (82), p ( ω i ) ≥ i , P ni =1 p ( ω i ) = 1 and with eq. (75), b ik a ik + b ik ≥ p ( ω i ). So, by lemma 3, P ni =1 g ik ( ω ) p ( ω i ) ≥ g ik . Therefore, p is a feasible solution of eqs. (76), (77) and (78) and by [13, p. 175, ll. 10–13], D avoids sure loss. (cid:3) VA L UA T I N G B E TT I N G O DD S AN D F R EE C O U P O N S U S I N G D E S I R A B I L I T Y Appendix D. Betting odds on the winner of the European Football Championship 2016
Countries bookmakers B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t B e t France 3 3 3 3 3 11/4 3 16/5 3 16/5 16/5 3 3 3 16/5 3 10/3 16/5 16/5 16/5 3 16/5 3 3 3 3 3Germany 4 4 9/2 4 9/2 4 4 9/2 10/3 9/2 9/2 9/2 4 7/2 4 9/2 9/2 19/5 9/2 15/4 9/2 19/5 9/2 4 9/2 22/5 23/5Spain 5 5 9/2 5 9/2 5 5 9/2 5 9/2 5 9/2 5 5 5 5 9/2 5 9/2 5 9/2 5 5 24/5 24/5 5 5England 17/2 9 9 8 9 8 9 8 9 9 8 9 8 8 9 9 9 17/2 9 9 9 17/2 8 8 17/2 43/5 9Belgium 11 10 10 10 10 11 10 11 10 10 10 10 11 10 11 11 11 9 10 9 10 9 10 10 54/5 53/5 57/5Italy 16 16 18 16 18 16 16 16 16 16 16 18 18 16 18 16 18 16 16 14 16 16 14 17 18 89/5 91/5Portugal 18 18 18 18 18 18 18 14 20 17 18 18 14 18 12 18 20 15 17 18 17 15 18 268/17 88/5 92/5 91/5Croatia 25 25 22 25 22 25 25 25 20 25 25 22 25 25 25 25 25 25 25 25 25 25 22 25 26 24 27Austria 40 40 33 33 33 40 40 40 33 40 40 40 33 33 28 40 33 40 40 33 40 40 33 40 45 43 45Poland 50 50 50 50 50 40 40 50 50 50 40 50 50 50 50 40 50 45 50 40 50 45 50 50 47 48 50Switzerland 66 40 66 50 66 66 50 50 66 66 50 50 50 50 50 66 66 66 66 66 66 66 50 60 66 65 64Russia 66 66 80 66 80 80 66 66 66 80 66 50 66 66 50 66 66 66 80 66 80 66 50 66 85 84 79Turkey 80 80 80 80 80 80 80 66 80 80 66 66 66 66 80 80 66 80 80 80 80 80 80 80 94 92 89Wales 80 80 80 80 80 80 66 80 100 80 80 66 66 66 100 66 66 80 80 80 80 80 60 80 89 81 89Ukraine 100 66 80 80 80 80 80 66 80 80 80 50 80 80 50 50 80 90 80 100 80 90 80 100 94 86 89Sweden 100 80 100 80 100 100 100 100 100 100 100 100 100 80 66 100 100 100 100 100 100 100 80 100 104 90 99Czech Rep 125 100 125 80 125 100 100 125 80 100 100 125 66 100 100 125 100 100 100 100 100 100 100 100 132 135 99Slovakia 150 100 150 150 150 150 150 150 100 100 150 150 150 150 100 125 125 187/2 100 150 100 100 125 150 142 143 119Rep of Ireland 150 150 150 150 150 125 150 100 150 150 150 150 100 125 100 125 125 349/4 150 150 150 112 125 150 170 156 149Iceland 100 150 100 100 100 100 100 150 80 100 80 150 80 100 100 150 100 110 100 100 100 110 60 100 180 179 149Romania 200 100 150 125 150 200 200 125 150 260 150 150 150 150 80 200 150 399/4 260 200 260 287/4 125 200 275 256 238N Ireland 350 250 400 400 400 350 350 300 300 400 300 300 250 250 300 250 400 359/4 400 300 400 120 350 400 389 377 376Hungary 350 250 400 200 400 350 350 400 350 400 250 300 250 200 200 250 250 359/4 400 250 400 359/4 250 350 541 566 79Albania 500 250 500 400 500 350 500 400 500 500 250 200 300 250 300 400 500 363/4 500 300 500 177/4 400 500 531 513 495
Table 9.
Durham University, Department of Mathematical Sciences, UK
E-mail address : [email protected] Durham University, Department of Mathematical Sciences, UK
E-mail address : [email protected] Durham University, Department of Mathematical Sciences, UK
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