Arb-mirage: Exploring the Extent to Which Apparent Inefficiency in Betting Markets is an Illusion

Arb-mirage: Exploring the Extent to Which Apparent Inefficiency in Betting Markets is an Illusion December 5, 2014 Andrew Grant a,, Johnnie E. V. Johnson b, and Tasos Oikonomidis b a University of Sydney Business School b Centre for Risk Research, School of Management, University of Southampton * Corresponding author,. Tel: +61 (2) 9036 6238. Fax: +61 (2) 9351 6451. Email addressses: andrew.grant@sydney.edu.au ; J.E.Johnson@soton.ac.uk ; tasos33bc@yahoo.co.uk 1

Arb-mirage: Exploring the Extent to Which Apparent Inefficiency in Betting Markets is an Illusion Abstract This paper explores the claim made in previous studies that the football betting market is weak form inefficient, to the extent that prices offered by competing bookmakers enable arbitrage to take place. We dichotomise bookmakers into those that change odds frequently and accept bets from sophisticated bettors (book-balancers) and those that do not change odds frequently and seek to avoid sophisticated clients (position-takers). Using data from European football, arbitrage opportunities are created by betting at the highest odds on each outcome across a pool of six bookmakers (three of each type). More than 50% of arbitrage opportunities arise from betting on favoured teams with the position-taking bookmaker, and hedging with the book-balancing bookmaker. Position-taking bookmakers set less efficient prices than book-balancers in order to attract uninformed order flow, who prefer to bet on favourites. We argue that these potential opportunities are not likely to be exploitable in practice as position-taking bookmakers are likely to restrict the activities of informed bettors.

Arb-Mirage: Are Inefficient Betting Markets an Illusion? 1 Introduction Over-the-counter (OTC) markets do not use centralised mechanisms such as auctions, specialists, or accessible limit order books to aggregate bids and offers (Duffie, 2012). Participants in OTC markets, often with varying degrees of information and bargaining power, negotiate terms privately, and may be unaware of (or unable to access) other prices available in the marketplace. The 2007-09 financial crisis has brought OTC markets in financial derivatives, such as credit default swaps, and collateralised debt obligations, under close scrutiny. The opaqueness of such markets deterred investors from trading, as the price discovery process was hampered with the severe difficulties in obtaining liquidity. The potential for dealers to suffer losses, or refrain from trade because of the unknown risks faced by counterparties of other dealers (Brunnermeier, 2009) appears to have contributed to the destabilisation of the market during the financial crisis. Policy discussions are ongoing regarding trade clearing, transparency, and information dissemination in over-the-counter financial markets. Bookmaker-driven betting markets (as opposed to betting exchanges or totalisator systems) provide a useful analogue to OTC markets for financial derivatives, or dealers on exchanges (Croxson and Reade, 2011). For example, the purchaser of credit protection in a credit default swap transaction is wagering that a corporate bond will be downgraded (a credit event will occur). The counterparty to this transaction (the credit protection seller) is analogous to the bookmaker. Our paper utilises the field experiment that is provided by betting markets to analyse the impact of trader sophistication and bargaining power on market efficiency in over-the-counter markets. There is a strong heterogeneity in the behaviour of bookmakers in the market for European football (soccer) betting. We characterise bookmakers as either position-takers or book-balancers, based on their strategic operations. Position-taking bookmakers, such as William Hill or Ladbrokes cater mainly to retail clients, and operate relatively high-transaction cost, low turnover strategies. Position-taking bookmakers actively seek to exclude sophisticated customers (Franck, Verbeek, and Nüesch, 2013) because they may maintain unbalanced inventory positions (their profits will depend on match outcomes). Book balancers, on the other hand, including SBOBet and Pinnacle, operate almost exclusively online, operate a high-turnover, low-transaction cost strategy. Most importantly, book-balancers 3

Grant, Johnson, and Oikonomidis place fewer restrictions on the actions of sophisticated counterparties, and actively seek to hedge their inventory position by adjusting their prices. In this paper, we investigate the effect of the different types of bookmaker operation on the efficiency of the globalised European football betting market. With the advent of competing betting exchanges such as Betfair, where bettors can either back or lay (bet for or against) a match outcome, a bettor has a choice to trade through either a dealer or double-auction exchange. Parallels may be drawn with financial markets operating a dual mechanism, such as NASDAQ dealers offering competing quotes on NYSE stocks. Interestingly, whilst betting exchanges tend to provide more accurate forecasts of match outcomes, and lower transaction costs than position-taking bookmakers (e.g. Franck et al, 2010), the dealer market coexists and has, indeed, thrived (Croxson and Reade, 2011, 2014), likely for liquidity considerations. Bookbalancing bookmakers, on the other hand, operate with transaction costs of a similar magnitude to the betting exchange, but with higher turnover than position-takers (and hence higher liquidity than the betting exchange in general). Consistent with financial markets, betting exchanges also offer a greater degree of pre-trade transparency than bookmakers; the three best bid and offer quotes and depth are typically available on the betting exchange. In contrast, position-taking bookmakers simply restrict large bets, whilst book-balancers typically offer some indication of the maximum stake size they will accept. A summary of the defining characteristics of position-taking bookmakers, book-balancing bookmakers, and betting exchanges is provided in Table 1. Table 1. Characteristics of Position-Taking Bookmakers, Book-Balancing Bookmakers, and Betting Exchanges. Characteristic Position-Takers Book-Balancers Betting Exchange Example Firm Ladbrokes SBOBet Betfair Domain Physical World/Online Online Only Online Only Incorporated U.K./Europe Asia U.K./Europe Clientele Retail Only Unrestricted Unrestricted Trading Cost (Overround) High Low Low Liquidity at Best Bid Low High Medium Freq. of Odds Changes Low High High Inventory Risk High Low N/A Pre-Trade Transparency Quotes Only Depth at Best Quote Depth at Best Three Quotes Volume Product 1 2 (Home/Draw/Away) Asian Handicap 1 2 (Home/Draw/Away) Several studies have explored the efficiency of betting markets by searching for arbitrage opportunities which may arise from disparate prices in alternate trading venues. This typically leads 4

Arb-Mirage: Are Inefficient Betting Markets an Illusion? to the construction of a synthetic Dutch Book, wherein a bettor can lock in a risk-free profit, regardless of an event s outcome, by betting at the highest individual-outcome odds for each outcome across a panel of bookmakers. As betting markets are increasingly accessible online, it is often suggested that the identification of arbitrage opportunities indicates an inefficient market. In this paper we suggest that arbitrage opportunities in betting markets are illusory because they usually require bets with retail-focused position-taking bookmakers. These bookmakers are inaccessible to most sophisticated gamblers and actively manage their portfolio of clients by restricting these supposed informed bettors (Franck, Verbeek, and Nüesch, 2013). The remaining counterparties for the position-taking bookmaker bet smaller amounts with higher margins, and are actively targeted by advertising and sign-up benefits including overly-generous promotional odds. In order to identify sophisticated gamblers, position-taking bookmakers often examine the relative pricing of their bets; bettors who mainly stake when the odds at the position-taker are relatively high can be easily identified and excluded. 1 Our tests consider whether the existence of cross-market arbitrage opportunities is the product of structural differences between demand and supply driven markets, and subsequently explore the impact of intentional inefficient pricing by position-taking bookmakers. The objective in their odds setting is not necessarily to solely reflect true outcome probabilities, rather the odds also reflect the bookmakers marketing strategies which are designed to acquire and retain customers. In this study, we demonstrate that cross-bookmaker arbitrage opportunities arise in approximately 25% of all football matches played in the 2012-13 season in six major European leagues. Three bookmakers are identified as book-balancers, who allow large trades and operate a lowmargin, high-turnover strategy. The remaining four bookmakers in our sample are identified as position-takers. We use linear programming techniques to construct risk-free portfolios from thirteen available bets for any given football match which are generally available in betting markets. These are standard 1 2 offers (where the 1,, and 2 reflect bets on Home Team, Draw, and Away Team, respectively), as well as the ten Asian Handicap bets with payoffs that can be constructed from the match outcomes (that is, do not require goal margins). In such Asian Handicaps, football bets typically become two-outcome propositions, with a decisive result being required for 1 In this respect a further analogy can be drawn between betting markets and financial markets; position-taking market makers provide liquidity similar to internalized trades on proprietary dark pools whereas book-balancers are more like internalized trades through the broker-dealer (see Zhu (2014) for a review of dark pools). 5

Grant, Johnson, and Oikonomidis bets to payoff, or stakes refunded in the case of a draw. 2 The focus of our study is to examine the source of arbitrage opportunities, and therefore determine whether a bettor would be able to operationalise such a betting strategy. We find that the large majority (84%) of apparent arbitrage opportunities would require a bettor to place a stake with both the position-taking bookmaker and book-balancing bookmaker, rather than within bookmakers of the same type. Among the arbitrage portfolios requiring bets across bookmaker types, bets on the most likely outcome ( favourite team) are more likely to be placed with the position-taking bookmaker. This arises because position-taking bookmakers are likely to face greater competition for bets on favoured teams (which are more popular among the gambling public), and may be willing to accept losses by holding positions against such propositions in order to maintain an active portfolio of clients (Forrest and Simmons, 2008; Franck, Verbeek, and Nüesch, 2011). However, because position- takers restrict the activities of sophisticated bettors, it is doubtful that such as strategy could be implemented successfully. The profitability of the apparent arbitrage strategies may be attributed to bets against the position-taking bookmaker. Placing unit-stake ($1) bets on all components of arbitrage portfolios requiring bets against position takers realised average profits of $0.16 per bet. Conversely, unitstake bets on portfolio constituents at the book-balancing bookmaker returned profits of $0.024 per bet, in the region of transaction costs. We demonstrate that odds from book-balancing bookmakers exhibit a lower degree of favourite-longshot bias than those of the position-taking bookmakers. Thus, in order to successfully execute arbitrage strategies, a bettor would need continuous access to position-taking bookmakers, who actively eschew trades from informed bettors (including arbitrageurs). This study contributes to existing literature in a number of ways. It is the first to analyse fixed-odds arbitrage betting in an operationally realistic manner; the offered prices from multiple bookmakers being measured contemporaneously and near to kick-off when markets have enough depth to provide meaningful economic returns to bettors. We can be assured that the panel of odds for a match would have been available to a bettor by virtue of the fact that they are collected simultaneously from bookmakers websites in the two hours prior to kickoff, and we are able to demonstrate that depth available at a book-balancing bookmaker increases close to kickoff. By 2 We provide more details on the exact payoffs to Asian Handicap bets later. 6

Arb-Mirage: Are Inefficient Betting Markets an Illusion? contrast, previous studies (e.g. Pope and Peel, 1989; Deschamps and Gergaud, 2007; Vlastakis et al, 2009) exploring cross-bookmaker arbitrage have assumed that odds from multiple bookmakers would have been available simultaneously and do not provide insight into market liquidity. Our results indicate that, whilst arbitrage profits are theoretically plausible, the institutional features of bookmaker-driven betting markets make implementation very difficult. Our results also have implications for financial markets generally. The fact that dealer-based markets coexist (and thrive) alongside the betting exchange suggests that increased transparency does not generally drive liquidity, even in relatively active products such as vanilla European football bets. The active exclusion of sophisticated traders by position-taking bookmakers does not seemingly affect ability to function in a separating equilibrium, alongside the lower-cost, more liquid book-balancing firms. Promotional pricing, in order to identify potential arbitrageurs, appears to be a useful strategy to retain only uninformed clients at the position-taking firms, and thus maximise profitability. 2 Background 2.1 Arbitrage and Efficiency in Financial and Betting Markets The notion of the Efficient Market Hypothesis (EMH) of Fama (1970) may be justified by the fact that rational arbitrageurs would drive temporary deviations in prices towards efficient benchmarks. Prices will therefore reflect fundamental values in efficient markets, providing that the arbitrageurs are willing and able to trade sufficiently to impact market values (Friedman, 1953). Asset prices may not reflect fundamental values when arbitrageurs face constraints or limitations (e.g. Pontiff, 1996; Shleifer and Vishny, 1997), a phenomenon known as limits to arbitrage (see Gromb and Vayanos (2010) for a review of this area). Restrictions faced by arbitrageurs are therefore major contributing factors to various financial market anomalies, some of which have persisted following their discovery. 3 As discussed in Akbas et al (2014) and McLean and Pontiff (2014), the fact that there is at least some persistence in anomalies suggests that data mining or statistical bias does not drive the returns. Moreover, the 3 See, for example, Bernard and Thomas (1989) for out-of-sample evidence on post-earnings announcement drift, and Rouwenhorst (1998) for international evidence on momentum, which are two of the most widely studied anomalies. 7

Grant, Johnson, and Oikonomidis most persistent anomalies have arisen in difficult-to-arbitrage stocks (McLean and Pontiff, 2014) and it is difficult to attribute many of the anomalies to risk-based explanations. A number of studies have examined betting markets as a cleaner test of market efficiency (see Sauer (1998) and Vaughan Williams (1999, 2005) for literature reviews of gambling markets generally, whilst Oikonomidis and Johnson (2011) review football betting market efficiency). As noted by Thaler and Ziemba (1988), betting markets offer advantages over financial markets for testing efficiency, as the objective value of assets are revealed with certainty, and in a timely fashion (i.e. when the game is finalised). Because stocks are infinitely-lived, their price depends on the present value of future cash flows, and on the price someone would be willing to pay tomorrow. 4 Hence, tests of financial market efficiency assume an equilibrium model that defines normal security returns (Croxson and Reade, 2014); efficiency may be rejected on the basis that the market is inefficient or the proposed model is incorrect (i.e. the joint hypothesis problem). Moreover, the structure of betting markets resembles that of financial markets in terms of the varying levels of sophistication of traders, who risk real-world money on the uncertain outcome of future events. Betting markets therefore may provide more generalisable results than those from laboratory experiments, in which subjects may be inexperienced traders or ineffective motivation (Levitt and List, 2007). Along the lines set by Fama (1970), Sauer (1998) indicates that a betting market is considered efficient if opportunities it is not possible to generate abnormal returns (strong test), or if differential returns are available to the bettor simply by placing stakes at different odds, such as on favourites (weak test). In general, the main determinant of betting market efficiency is the degree to which the market odds reflect the unknown, true probabilities of event outcomes over a large sample. This is analogous to efficiency in wider financial markets, where efficiency is determined by the degree to which market prices reflect fundamental values. However, as discussed below, studies which explore the EMH should consider the structural idiosyncrasies of the betting market, so that successful inferences concerning market inefficiency are drawn. Prior studies of arbitrage prospects in betting markets show mixed results. Some studies of horserace betting markets find against the existence of arbitrage (e.g. Adams et al, 2002; Gramm 4 The problem of stock prices potentially deviating from fundamental value similarly affects tests of efficiency in options market, even though they are short-lived. 8

Arb-Mirage: Are Inefficient Betting Markets an Illusion? and Owens, 2005), whilst other find there are possible arbitrage opportunities (e.g. Willis, 1964; Hausch and Ziemba, 1990; Edelman and OBrian, 2004). Horse racing employs pari-mutuel (totalisator) wagering markets. As such, arbitrage cannot be categorised as risk-free, since the effective odds that the bettor eventually secures cannot be certain at the exact time of the bet, unless that bet is the last one for that race. These cases should be classified as quasi-arbitrage (Paton and Vaughan Williams, 2005) or risk-arbitrage opportunities (Lane and Ziemba, 2004). In such cases, the dispersion of prices may lead to a betting opportunity, but outcome-independent profits are not guaranteed. The possibility of zero risk arbitrage has also been explored in the fixed odds football betting market, typically examining the extent to which the differences in the odds quoted by different bookmakers are adequate to guarantee profitable, fully hedged positions. In an early study, Pope and Peel (1989) show that arbitrage opportunities exist due to price dispersion across bookmakers. However, later research suggests that the degree of coordination between bookmakers has increased, possibly due to the emergence of professional arbitrageurs (Dixon and Pope, 2004). The latter finding is in line with further studies (Deschamps and Gergaud, 2007; Luckner and Weinhardt, 2008; Deschamps, 2008; Vlastakis, Dotsis and Markellos, 2009; Spann and Skiera, 2009; Franck, Verbeek and Nüesch, 2010), which show that bookmakers prices are fairly well aligned (along with relatively high transaction costs) and as a result cross-bookmaker arbitrage opportunities rarely arise in the major football betting markets. Studies that extend the investigation of price dispersion into a less homogeneous population of bookmakers typically reach different conclusions. In an interesting empirical study, Marshall (2009) analyses data from 50 different bookmakers originating from a variety of jurisdictions, covering a range of sports (including football) between January 2003 and December 2005. He finds that 19,882 arbitrage opportunities existed overall in this period and that they lasted 15.75 minutes on average. 5 In the study most closely related to ours, Franck, Verbeek and Nüesch (2013) analyse bookmakers odds in parallel to betting exchange odds for a sample of 12, 782 European football matches 5 Given that Marshall obtained such data from a company supplying software that explores arbitrage opportunities, 15.75 minutes seems to be a considerable amount of time for the arbitrage opportunities to take to disappear, as one would expect the automation offered by such software to lead to orders being immediately placed against outlying odds by its users. This should lead to an almost immediate correction (i.e. convergence of the outlying prices towards the market mean). 9

Grant, Johnson, and Oikonomidis over the seven-year period from 2004-05 to 2010-11. They find that cross-bookmaker arbitrage opportunities exist in 0.8% of matches in their sample, which includes only position-taking European bookmakers. When considering the bookmaker odds in parallel to the betting exchange Betfair, Franck et al. (2013) show that the proportion of matches in which arbitrage opportunities arise increases to 19.2%. Such a strategy involves taking a long position at the bookmaker, and laying the same position at the betting exchange. 6 Profits from the inter-market arbitrage portfolios were almost exclusively against the bookmakers; in arbitrage portfolios profits against the bookmaker were around 7%, while profits at the betting exchange were very close to zero. Conclusions regarding the existence of arbitrage opportunities depend upon the choice of market under investigation. Arbitrage opportunities arise more frequently between market structures than within market structures (Edelman and O Brian, 2004; Franck, Verbeek, and Nüesch, 2013). However, to date most studies analysing arbitrage in betting markets have considered bookmakers operating as position-takers. Levitt (2004), for example, notes that position-taking bookmakers strategically set odds in a manner to profit from the biases of uninformed bettors. This study seeks to uncover the extent to which this price setting mechanism affects market efficiency when alternative (book-balancing) bookmakers, who do not seek to exploit trader biases, operate in parallel. 2.2 The Behaviour of Bookmakers Levitt (2004) argues that betting markets are organized very differently from financial markets. As the main providers of liquidity, bookmakers take large positions against their customers rather than necessarily matching sellers with buyers and simply earning the commission from the overround (spread). According to this position-taking model, bookmakers profits are related to match outcomes. This approach to bookmaking is examined in a number of other studies (e.g. Kuypers, 2000; Paul and Weinbach 2007, 2008; Humphreys, 2011). Franck et al. (2013) suggest that bookmakers do not only attempt to maximize their profit by taking positions against their customers, but also effectively choose the punters against whom they take such positions. They achieve this by monitoring client trades and restricting service to the ones profiled as potentially skilled. In this 6 Long only arbitrage portfolios employing the betting exchange and bookmakers in the creation of a synthetic Dutch book were also considered in the study by Franck, Verbeek, and Nüesch (2013), but arose in only 5.0% of matches. 10

Arb-Mirage: Are Inefficient Betting Markets an Illusion? context, bookmakers may occasionally publish inefficient odds that are likely to attract customers, safe in the knowledge that they can eliminate those who place the majority of their bets at these promotional inefficient prices. The damage to the bookmaker due to the setting of theoretically inefficient odds is therefore minimized, and is potentially lower in comparison to the gain in the size of the customer base. An alternative view is that a bookmaker s objective is to balance their books, and as a result, secure profit independent of the event s outcome (Magee, 1990; Woodland and Woodland 1991, Hodges, Lin, and Liu, 2013). A bookmaker operating in this manner changes his odds frequently in order to account for inventory imbalance. Such bookmakers act as uninformed market makers and essentially set up an over-the-counter market. Holding zero-book these market-making bookmakers act as though they are infinitely risk-averse (Fingleton and Waldron, 1999) and could plausibly charge lower transaction commissions, due to the absence of adverse selection costs. As a comparison, at betting exchanges, a bettor s liquidity is supplied by other bettors willing to take the opposite position. Franck et al. (2013) note that bookmakers provide the equivalent advantage in betting exchanges that dealers offer in auction markets: guaranteed liquidity (Madhavan, 2000). Moreover, when it comes to market making bookmakers, problems related to discrimination against skilled bettors are less likely to arise, since their model is based on the maximisation of volume rather on successful positions (see Forrest, 2012 for more details of the book-balancers model). In football betting, despite the fact that position-taking and book-balancing bookmakers coexist, the literature has almost exclusively focused on analysing odds offered by the former. This is surprising as the economic significance of the latter is probably greater, at least in terms of the volumes wagered (Forrest, 2012). Given the significant differences in the structure of the bookmaker s models, arbitrage opportunities between the two types of bookmakers may well be regarded as inter-market opportunities, similar to those between a bookmaker and tote, or bookmaker and betting exchange. Betting exchange odds are more accurate predictors of event outcomes compared to bookmaker odds (Smith, Paton, and Vaughan Williams 2006, 2009; Spann and Skiera, 2009), which suggests that arbitrage opportunities emerge due to the bookmakers inefficient pricing. Thus, (Franck et al., 2013, p.320) explain that it would be easy for bookmakers to align their odds with those offered 11

Grant, Johnson, and Oikonomidis in betting exchanges. The fact that they tend not to do so indicates that these inefficiencies are intentional due to structural differences between the markets. This is consistent with their theory that bookmakers do not set their odds in order to maximize the profit per game, but their long term overall profit (arising from a growing customer base). 7 The theory presented by Franck et al. (2013) is innovative as it links odds-setting with the bookmaker s option to withhold service from those clients it believes to be informed. 8 This presents a distinction between betting and financial markets, which should be accounted for when the efficiency of odds published by bookmakers is investigated. 9 Due to the nature of arbitrage, which requires that different prices must be available at exactly the same time, there are a number of technical details upon which our study can be compared to that of Franck et al. (2013). Firstly, the bookmakers odds which are utilised by Franck et al. (2013) bear no time-stamp; rather, the time of the offer is assumed, and odds are assumed to be constant for a given time interval. This is not always true, even for position-taking bookmakers, and as a result the matched offers between the betting exchange and the bookmakers are not guaranteed to have coexisted at the time of the bet. Secondly, the matched offers between the exchange and the bookmakers can be measured up to 2 days prior to kick-off, when the amount of money that can be staked in a betting exchange is very low, suggesting that even if arbitrage opportunities exist, they may not be economically meaningful. Thirdly, Franck et al. (2013) examine the exchange Betfair, which levies a commission on winning customers (referred to as premium charges ), the implication of this for the generation of a risk-free portfolio is ignored. 10 Fourthly, even though the volume that one can stake in a betting exchange 7 A related issue is how bookmakers manage their inventory. Traditional inventory models such as Ho and Stoll (1983) and Froot and Stein (1998) suggest that securities dealers should manage inventory at the portfolio level (some inventory imbalances will be hedged naturally). However, dealers in U.K. gilts (treasury bonds) appear to manage inventory on an individual security basis, rather than at the portfolio level, as pointed out in Naik and Yadav (2003). The ability of bookmakers to hedge is reflected in their ability to trade with other bookmakers. Forrest (2012) suggests that position-taking bookmakers may use the book-balancers for hedging purposes. 8 An example of the restriction notices from Bet365 and William Hill, among others can be found at http: //www.the-secret-system.com/bookmakers-shutdown-messages.html. Position taking bookmakers refer to the use of restrictions as a commercial deicsion. 9 Proprietary dark pools (equity trading services that do not publicly display orders) offered by firms such as Getco and Knight Capital trade on principal accounts, and may exclude sophisticated, or informed counterparties. Because they are relatively opaque in their execution services, and do not guarantee execution (especially for informed investors), they present an interesting analogue to position-taking bookmakers. See Zhu (2014) for further details on proprietary dark pools. 10 Betfair may withhold up to 60% out of winning bettors profits (see the website http://www.betfair.com/www/ GBR/en/aboutUs/Betfair.Charges/) 12

Arb-Mirage: Are Inefficient Betting Markets an Illusion? increases as the kick-off time approaches, it is shown below that there is significant variation in the size of the stakes that can be placed across games. This harms the homogeneity of the sample, as several apparent arbitrage opportunities may bear no economic significance if only small wagers are possible. Consequently, it is important to test the proposition suggested by Franck et al. (2013) on a data set that does not exhibit the limitations outlined above, in order to explore whether robust empirical evidence supports their theoretical proposition. We explore whether arbitrage opportunities exist between book balancing and position taking bookmakers odds and investigate the roots of any observed price dispersion. Our tests consider whether the existence of cross-market arbitrage opportunities is the product of structural differences between demand and supply driven markets, and subsequently explore the impact of intentional inefficient pricing by position-taking bookmakers. The objective in their odds setting is not necessarily to solely reflect true outcome probabilities, rather the odds also reflect the bookmakers marketing strategies which are designed to acquire and retain customers. However, we argue, in contrast to Franck et al. (2013), that this approach to price setting cannot be generalized to all bookmakers. Rather, it only applies to those falling under the group of position takers. This could also explain the findings of Marshall (2009), who, as opposed to Franck et al. (2013), identifies the presence of arbitrage opportunities within the bookmaker market, but does not investigate the nature of the bookmakers where these opportunities usually arise. The proposition tested in this paper is that arbitrage opportunities are most likely to arise between position takers and book-balancers, and not within odds from bookmakers of the same type. In addition, we argue that the investigation of the EMH when using odds provided by position taking bookmakers may lead to biased conclusions. Specifically, as suggested by Franck et al. (2013), such market operators may intentionally set inefficient prices as a marketing strategy to attract customers. However, if the operating strategies of these bookmakers ensure that these prices cannot be systematically exploited by informed bettors then any observed arbitrage opportunities are merely an illusion. On the supply side, this policy would lead to a deliberate inefficient pricing strategy by profit-maximizing bookmakers over their entire customer base due to the elimination of informed bettors. The cost of accepting a relatively small number of bets against inefficient odds is likely to be low, compared to the benefit of acquiring and preserving customers, who might consider alternative betting outlets if attractive odds were never offered by the bookmaker. Given 13

Grant, Johnson, and Oikonomidis that position-taking bookmakers are likely to encompass such marketing related objectives in their odds setting, it could be claimed that seemingly inefficient odds may actually be very efficient after considering the objectives of the odds-setters. Consequently, any conclusions regarding market efficiency based solely on the advertised position-taking bookmakers odds could be unreliable. However, such complications in the assessment of market efficiency should not exist when employing the odds of book balancing bookmakers, since their prices should be a more accurate reflection of betting volumes staked in the market. Odds arising from this market constitute more appropriate data for the testing of the EMH in the football betting markets. 3 Hypothesis Development There are two types of bookmaker, position-takers and book-balancers, simultaneously operating in the betting market, pursuing differing objectives. It is expected there are occasions when prices in these markets are sufficiently disparate for arbitrage to appear possible. Such instances are expected to mainly arise between bookmakers from these different groups rather than between bookmakers from the same group. However, because position taking bookmakers effectively prevent skilled traders from exploiting these opportunities, such arbitrage is effectively non exploitable in the long run. Hence, we refer to these instances as arb-mirage. On the one hand, there are the book-balancing bookmakers, who attempt to maximize the volume traded on their books and move their odds in a way so that their books are fairly balanced; thereby minimizing their risk. The odds offered by these bookmakers are expected to be mainly driven by smart money 11 and to be well calibrated, since opportunities in this market are exploitable. Consequently, mis-pricings by these bookmakers are expected to be corrected, according to the EMH. On the other hand the position-taking bookmakers, whose objective seems to be to maximize their customer base (Franck et al, 2013) rather than the expected profit per game appear to operate policies to deter or prevent skilled punters from exploiting any mis-pricing. As a result of this policy the position-taking bookmakers may use their odds as a marketing tool to attract and retain customers (Marshall, 2009; Franck et al, 2013) and therefore, occasionally (intentionally) accept bets exhibiting a negative expected value. This arises because it may be beneficial to 11 For a description of a book-balancer s model see http://www.pinnaclesports.com/betting-promotions/ winners-welcome.aspx?ico=home&icl=box3 14

Arb-Mirage: Are Inefficient Betting Markets an Illusion? offer these prices to attract clients, believing they can prevent skilled traders from exploiting any inaccuracies in their pricing. Due to the desire to maintain zero-book, the frequency of price changes at book-balancing bookmakers is greater than that of position-takers. By contrast, the position takers rely on their higher over-round in order to allow greater stability in their odds, as the higher over-round provides a cushion against (possibly deliberate) inaccurate forecasts of match outcomes. Price stability also allows position-taking bookmakers retain promotional odds, and are particularly important at physical world betting shops. Due to the more informed nature of book-balancers clients, the volumes traded on particular outcomes are likely to be informative. Hence, any movement of odds of position-taking bookmakers should not necessarily be regarded as the product of arbitrageurs actions, since those are unlikely to have access to position-taking bookmakers (Franck et al, 2013). Moreover, given the existence of several arbitrage software platforms (e.g. Marshall, 2009) if position takers were moving their odds due to arbitrageurs actions, arb-mirage opportunities should barely be visible. In this interactive environment, in which relatively stable and frequently moving prices coexist, instances of significant price dispersion, leading to arb-mirage are anticipated. However, given the objectives of the two types of bookmakers, we expect arb-mirage to occur across bookmakers type (i.e. between position-takers and book-balancers). These apparent arbitrage opportunities will effectively reflect the deviation of position-takers odds from the outcomes objective probabilities, or at least from the probabilities incorporated in the odds of the book-balancing bookmakers. Consequently, the consideration of the less efficient position takers odds alongside the book-balancers odds is expected to lead to a significantly greater number of instances of arb-mirage. In order to explore the validity of the proposition suggested above, we test the following three related hypotheses: H1: There exist instances where the price dispersion in the betting market is adequate to generate seemingly risk-free opportunities for bettors to profit by simultaneously betting with different bookmakers on alternative outcomes related to the same event. Levitt (2004) studied trading volume from a major position-taking bookmaker, and showed that favourites constitute more popular bets compared to longshots. Bookmakers therefore face net exposure to favourites. Likewise, Forrest and Simmons (2008) and Franck, Verbeek and Nüesch 15

Grant, Johnson, and Oikonomidis (2011) show that position-taking bookmakers offer better prices for bets on popular teams, in order to sustain competition and to build/maintain their customer base. Consequently, we expect that in most cases where apparent arbitrage opportunities exist, the position-taker will post the best offer for the favourite and a book balancing bookmaker will be posting the best offer for the longshot. To explore this further we test the following hypothesis: H2: Apparent arbitrage opportunities most commonly arise between book-balancers and positiontakers, by position-takers offering the highest odds for favourites and book-balancers offering the highest odds for the longshots. Based on Franck et al. s (2013) finding that arbitrage profits between the betting exchange and position-taking bookmakers were generally earned at the expense of bookmakers, we expect that apparent arbitrage profits are earned at the expense of the position-takers, not the book-balancers. Arbitrage opportunities arise as a result of position-takers not adjusting their odds quickly enough to incorporate price-informative trends signalled by informed money traded with book-balancing bookmakers. H3a: Position-takers suffer losses when an apparent arbitrage opportunity exists. The expected profit of bets placed against such bookmakers is higher compared to that placed against bookbalancers, when apparent arbitrage opportunities occur. We also expect that the odds offered by book-balancing bookmakers are more calibrated than those offered by position-taking bookmakers. In other words, we expect that book-balancers odd odds-implied probabilities are more reflective of the long-run occurrences of match outcomes. If position takers were solely interested in efficient estimation of event probabilities, they could simply adjust their odds to those of the book balancers. However, if it is shown that the position taking bookmakers do not adjust their odds in this way then this is supportive of Franck et al. s (2013) proposition that marketing considerations forming part of their odds-setting strategy. H3b: Book-balancers odds constitute more accurate predictors of event outcomes (compared with those of position-taking bookmakers). Finally, since according to H2, position-takers are more often expected to offer the highest odds for the favourite in the apparent arbitrage opportunities, it is likely that such bookmakers odds underestimate the favourite s winning probability. H3c: Book-balancers odds exhibit a lower degree of favourite-longshot bias than position- 16

Arb-Mirage: Are Inefficient Betting Markets an Illusion? takers odds. 4 Data and Methodology 4.1 Bookmaker Description and Data This study attempts to explore structural differences between position-taking and book-balancing bookmakers and therefore, the employment of data from major bookmakers representing both groups is essential. We classify Ladbrokes, William Hill, Bet365, and Stan James as positiontaking bookmakers, and IBCBet, Pinnacle, SBOBet and 188Bet as book-balancing bookmakers, based on consultation with bettors and from information available on bookmakers websites. Ladbrokes and William Hill were established in 1886 and 1934, respectively, and operate retail businesses with thousands of betting shops, mainly in the U.K. They also operate online, and their combined aggregate gross revenue exceeds 1 Billion. 12 Bet365 is a major UK-based online betting company founded in 2001, which achieved turnover of 8.5 Billion and gross profit of 422 Million in 2011. 13 Stan James is private company with a well-established brand name, operating mainly online, while owning 65 betting shops in the U.K. 14 One of the defining characteristics of position-taking bookmakers is their physical-world presence, which is important to each of these bookmakers. Annual reports for these bookmakers note that their profitability depends on match results. 15 We also collected data from four of the leading book-balancing bookmakers: SBOBet, IBCbet, 188Bet, and Pinnacle. The first three of these are leading Asian bookmakers, handling enormous volumes, allegedly far higher than traded by more traditional European bookmakers (Forrest, 2012). Pinnacle is also a major online operator, purportedly trading billions of dollars. Pinnacle are fairly open regarding their operation (unlike the other Asian bookmakers), which they describe on their website 16 as attempting to maximise trading volume while minimising exposure, using information 12 See William Hill (2013) Preliminary Results 2012 and Ladbrokes (2013) Preliminary Results for the Year Ended December 2012. 13 Source: http://www.publications.parliament.uk/pa/cm201213/cmselect/cmcumeds/writev/1554/ga104. htm. 14 Source: http://howtobet.net/sportsbook-review/stan-james. 15 For example, Ladbrokes 2013 annual report (p.23) notes Ladbrokes may experience significant losses as a result of a failure to determine accurately the odds in relation to any particular event. 16 See, inter alia, http://www.pinnaclesports.com/about-us.aspx, http://www.pinnaclesports.com/ betting-promotions/arbitrage-friendly, and http://www.pinnaclesports.com/betting-promotions/ 17

Grant, Johnson, and Oikonomidis arising from informed traders as a tool to set efficient odds. In particular, Pinnacle emphasize that they are friendly to arbitrageurs, as the expected value of a trade for them should not depend on the motives of the counterparty placing the stake (i.e. if the bettor is an arbitrageur or professional trader). The model described by Pinnacle also fits the operations of the three other main Asian bookmakers listed above and is fully aligned with the model of book -balancing bookmakers. Of importance is the lack of physical-world presence and a tendency to change odds frequently. In order to ensure potential arbitrage trades could have realistically been executed, we designed a data collection program to scrape odds information systematically from bookmakers websites. The scraping program was set up so that requests to bookmakers websites were conducted simultaneously, employing a forced a time-out to ensure a maximum discrepancy in odds collection of 30 seconds. 17 All odds were collected in a period within 2 hours from kick-off, when the staking limits reach their peak. We focused on the major European football leagues in order to ensure that the findings of the study carry economic significance, as the volumes traded in leagues of lower status are significantly smaller. We collected data, for the whole of the 2011-12 season for the 6 major leagues; the English Premier League, the German Bundesliga, the Italian Serie A, the Spanish La Liga, the French Ligue 1, and the Dutch Eredivisie. Overall, this resulted in a sample of 2,132 games, which we will refer to as the main sample. For each match, we collected odds information for several bookmaker products: the Asian Handicap (which we will occasionally shorten to AH), and home win, draw, and away win (also known as 1 2) markets. In Asian Handicap betting, one of the teams (usually the favourite) is given a goal-deficit (handicap) to overcome, the size of which is indicated by a negative number. The bettor s stake will pay off if the handicapped team wins the match by a greater margin than the handicap. A bet on the opposing team is successful providing they do not lose by a margin greater than the handicap. Fractional Asian handicaps include the draw outcome as relevant to the stake. For example, a team with a handicap of 0.5, starts with a half goal deficit, meaning that winners-welcome. 17 The cost of this was that several bookmakers occasionally failed to respond within the maximum allowed period. In these cases, we repeated the full request (i.e. for all bookmakers) three times in order to obtain a complete sample. In some cases, due to heavy load on bookmakers websites, some would remain unresponsive. In those cases, the odds of those who failed to respond were not considered, which may lead to a slight underestimation of the frequency of arbitrage opportunities overall. The timeout could be increased in future studies, however, this would risk the integrity of the results overall, as a higher time interval would increase the chance of odds of the quickest responding bookmaker changing until the response of the slowest bookmaker came back. The collection of data from bookmakers remains a difficult practice at best. 18

Arb-Mirage: Are Inefficient Betting Markets an Illusion? the team must win outright (that is, not draw) for the bet to pay off. A team which is handicapped +0.5 starts with a half goal advantage, so a bet on that team wins even if the outcome of the game is a draw. A bet on a team with a 0 handicap is refunded in the event of the draw, whereas a bet on team with a -0.25 (+0.25) handicap is considered as a half-bet on 0 handicap and a half-bet on -0.5 (+0.5). 18 The Asian Handicaps that are of interest for this study are those in the interval of -0.5 to +0.5, because none of these bets in this range are dependent on the actual number of goals scored. Hence, an arbitrage portfolio can potentially be formed by betting with the home win, draw, and away win (1 2) markets. For our second data set we collect odds from Betfair to assess the relative liquidity of betting exchange odds with bookmaker odds, at points both two days and two hours prior to kickoff. Prior football betting studies (e.g. Direr, 2013; Franck et al, 2013) have typically obtained bookmaker data from websites that store historical odds without timestamps, from time points up to two days prior to kick off. 19 Our data set allows us to accurately examine the relative liquidity in each type of market at points marked an exact time before kickoff. We aim to determine the economic significance of arbitrage opportunities by examining the relative liquidity available to an arbitrage trader on the betting exchange and at a book-balancing bookmaker. SBOBet reports the maximum stake a bettor could implement at any point in time. We use their transparent pre-trade liquidity for the most liquid Asian Handicap bet, as a comparison to the maximum bet size available at the best bid plus best offer on Betfair. Liquidity data was collected for a sample of 115 matches for the top six European leagues in our sample over the period 23/8/2013 until 1/9/2013. Our final set of data contains historical odds from six bookmakers (Ladbrokes, William Hill, Bet365, SBOBet, 188Bet, and Pinnacle) for all matches each of the six leagues in our sample, for an extended sample of 6,396 matches over the three seasons from 2009-10 to 2011-12. We obtained this dataset in a similar fashion to our main data, by scraping the bookmakers websites. This dataset is used to explore the relative incidence of favourite-longshot bias amongst the odds, or the degree of calibration between the odds-implied probabilities of match outcomes with the actual match outcomes. We obtained home win, draw, and away win odds for each match, at a time point within 18 We present a detailed example of the returns to each bet in the next section. 19 The most commonly used data source for football betting odds is www.football-data.co.uk. Other recent studies (Deschamps and Gergaud, 2007; Deschamps, 2008) have similarly used football betting data without timestamps. This is a reflection on a lack of data availability from bookmakers. 19

Grant, Johnson, and Oikonomidis two hours of match kickoff. We cannot use this entire sample (excluding 2011-12 data) to test the economic efficiency of arbitrage portfolios, as odds were not collected to ensure simultaneous execution of trades. However, the longer data period allows greater power in statistical tests of efficiency. 4.2 Methodology 4.2.1 Estimating the Frequency of Arbitrage Opportunities. In order to test H1 (existence of arbitrage opportunities between bookmaker types) we formulate a linear optimisation problem, where the objective is to decide the optimal distribution of stakes across different products in order to maximise the return. Arbitrage opportunities will be deemed to exist where such a return is positive and invariant across all match outcomes. Obviously, if there is not enough dispersion in the odds across the market to generate an arbitrage opportunity, there will not be a feasible solution to the problem. Providing there is sufficient dispersion, there will be a range of solutions that offer certain positive returns and the linear program will suggest the combination that offers the highest profit. The method below identifies how this problem is formulated and examined for a single game. This process is repeated for all 2,132 games in the sample. Let X j,k denote the vector of gross odds offered by bookmaker k, where for each game there are 1 j 13 products offered by the bookmaker (i.e. home win, home win with a -0.5 handicap, away win with a +0.5 handicap, etc). The gross return to each of the 13 bookmakers products is presented in Table 2, with products 1, 7, and 8 indicating odds from the 1 2 market, and the rest (with suffix AH) obtained from the Asian Handicap market. The odds in the Table 2 can by multiplied by stake size S to determine non-unit payouts. For example, if a bettor were to stake $5.00 on a Home Win on a +0.25 Handicap bet (j = 5) at gross odds of $2.20, and the match is drawn, their payoff (including the initial stake) would be 5 (1 + 0.5(2.20 1)) = 5 1.60 = $8.00. Alternatively, the $5 bet on the Away Win on a ( 0.25) Handicap bet (j = 10) would return $2.50, and the corresponding bet on the Away Win (0) Handicap bet (j = 11) returns the initial $5 to the bettor. In seeking a solution to the linear programming problem we first search for the highest odds across the set of k = 7 bookmakers in each market. This allows us to search for cases in which a synthetic Dutch book could potentially be constructed. We define the vector X max element-wise 20

Arb-Mirage: Are Inefficient Betting Markets an Illusion? Table 2. Gross return to a $1 stake for a single bookmaker on each potential match outcome for different types of bet, with odds vector (X 1,..., X 13 ) indicating the gross payoff for each corresponding market. The suffix AH indicates that the product is from the Asian Handicap market. Outcome Bookmaker Return if Return if Return if (j) Product Home Win Draw Away Win 1 Home X 1 0 0 2 Home (-0.5) AH X 2 0 0 3 Home (-0.25) AH X 3 0.5 0 4 Home (0) AH X 4 1 0 5 Home (+0.25) AH X 5 1 + 0.5(X 5 1) 0 6 Home (+0.5) AH X 6 X 6 0 7 Draw 0 X 7 0 8 Away 0 0 X 8 9 Away (-0.5) AH 0 0 X 9 10 Away (-0.25) AH 0 0.5 X 10 11 Away (0) AH 0 1 X 11 12 Away (+0.25) AH 0 1 + 0.5(X 12 1) X 12 13 Away (+0.5) AH 0 X 13 X 13 as X j,max = max k (X j,k ). In cases where bookmakers are tied for the highest odds, we retain all possible combinations of maximum prices. For example, if both Ladbrokes and William Hill were offering gross odds of $1.50 on a home win (j = 1) for a particular match, and this price was higher than all the other bookmakers prices for j = 1, we would retain two X max vectors, in order to not lose information for H1 and H2. Second, we aim to find the set of stakes that a bettor would place to best exploit potential arbitrage opportunities. Let S j be the bettor s allocated stake for each bet type j. The profit function Z = {Z homewin, Z draw, Z awaywin } for each possible match outcome can be defined from the following set of equations: 21

Grant, Johnson, and Oikonomidis 6 Z homewin = S j X j,max 13 j=1 j=1 S j (1) Z draw =S 7 X 7,max + 0.5(S 3 + S 10 ) + (S 4 + S 11 ) + 0.5(S 5 + S 12 + S 5 X 5,max + S 12 X 12,max ) (2) 13 +(S 6 + S 13 X 13,max ) 13 Z awaywin = S j X j,max 13 j=8 j=1 j=1 S j S j (3) To identify the best possible arbitrage opportunity one needs to find the distribution of stakes S that maximises the payoff for any of the three match outcomes, subject to a set of constraints. Hence, the optimisation identifies the distribution of stakes S that maximizes the payoff for any of the three match outcomes. The optimization routine can be written as Find optimal strategy S by varying S such that Z homewin is maximized (4) Subject to constraints Z homewin =Z draw (5) Z homewin =Z awaywin (6) Z homewin >0 (7) 13 j=1 S j =1 (8) S j 0 j (9) Due to the linear nature of the problem, the simplex algorithm can be used in order to maximise the objective function (Dantzig, 1951). Constraints (5) and (6) ensure that the selected combination of stakes leads to the same return independently of the outcome. Constraint (7) implies that for the solution to be acceptable, the net return should be positive. Constraint (8) requires that the sum of stakes should equal 1, so that each S j will represent the fraction of the available capital that should be staked on each bet type. Finally, constraint (9) requires that all stakes are positive. 22

Arb-Mirage: Are Inefficient Betting Markets an Illusion? The optimisation will fail to find a feasible solution in the event that arbitrage is not possible for the given set of bet types on a given game. If there is more than one feasible solution per game, we would select the bet with the highest return per outcome. By way of example, for the game between Mainz and Wolfsburg, played in Mainz s Coface Arena on 24/08/2013 at 14:30 BST, the bookmaker Pinnacle offered odds (X 1, X 2,..., X 13 ) of X = (2.73, 2.72, 2.36, 1.99, 1.70, 1.54, 3.58, 2.67, 2.67, 2.31, 1.95, 1.68, 1.52) (10) at 8:53 BST on the day of the game. Based solely on the offers of this bookmaker and maximising Z homewin, subject to constraints (5) to (9), no feasible solution is found. This indicates that these offers are internally consistent and no arbitrage opportunities are available (i.e. no Dutch book exists across the odds offered on the various types of bet offered by Pinnacle on this game). Suppose a different bookmaker offers odds of X 3 = 2.46 for Mainz on a Home Win (-0.25) Handicap (j = 3), X max becomes X max = (2.73, 2.72, 2.46, 1.99, 1.7, 1.54, 3.58, 2.67, 2.67, 2.31, 1.95, 1.68, 1.52) (11) and the maximisation routine yields an optimal solution of S = (0, 0, 0.4071, 0, 0, 0, 0.0794, 0, 0, 0, 0.5135, 0, 0). (12) In other words betting 40.71% of the bankroll on the Mainz (-0.25) Asian Handicap at odds of $2.46, 51.35% on Wolfsburg (0) Asian Handicap (j = 11) at odds of X max,11 = $1.95 and 7.94% of the bankroll on the draw (j = 7) at odds of X max,7 = $3.58, the bettor can secure a profit equal to 0.14% of the total investment, irrespective of the outcome of the game. A demonstration of the stake sizes and the return to each of the match outcomes is presented in Table 3. In order to test H1 we run this maximisation for each match in the sample. We consider the best odds for each offer in each match, in order to uncover whether cross-bookmaker arbitrage opportunities exist in the football betting market close to kick-off. 23

Grant, Johnson, and Oikonomidis Table 3. : Example of Arbitrage Opportunity from Linear Program for game between Mainz and Wolfsburg, played in Mainzs Coface Arena on the 24/08/2013 Outcome S X max Return if Return if Return if (j) Home Win Draw Away Win 1 0 2.73 0 0 0 2 0 2.72 0 0 0 3 0.4071 2.46 1.0014 0.2035 0 4 0 1.99 0 0 0 5 0 1.70 0 0 0 6 0 1.54 0 0 0 7 0.0794 3.58 0 0.2843 0 8 0 2.67 0 0 0 9 0 2.67 0 0 0 10 0 2.31 0 0 0 11 0.5135 1.95 0 0.5135 1.0014 12 0 1.68 0 0 0 13 0 1.52 0 0 0 SUM 1 1.0014 1.0014 1.0014 4.2.2 Identifying Favourites and Longshots by Bookmaker in the Arbitrage Portfolio. We now explore the methodology employed to test H2, that is, whether arbitrage opportunities are more likely to occur between position-taking and book-balancing bookmakers, rather than within a single type of bookmaker (that is, between position-takers or book-balancers only). We also examine whether the arbitrage portfolio is more likely to see bets on the favourite against the position-taking bookmaker. Matches for which an arbitrage opportunity was identified are isolated. We then compare the frequency of instances in which the position taking bookmaker offers the highest odds for the favourite (cf. the longshot ). The favourite is considered to be the outcome for which the lowest odds are offered in the 1 2 market by all the bookmakers. 20 We then define indicator variables D f and D l for each match i as D fi = 1 if the position-taking bookmaker offers the highest odds for the favourite on match i, and 0 otherwise, and D li = 1 if the position-taking bookmaker offers the highest odds on the longshot for match i, and 0 otherwise. Over the sample of n games in which an arbitrage opportunity arsies, we can calculate the relative frequency of cases where the best offer for the favourite was provided by a position-taking 20 We did not identify any cases in which bookmakers disagreed on the identity of the favourite in our arbitrage portfolios. 24

Arb-Mirage: Are Inefficient Betting Markets an Illusion? (book-balancing) bookmaker as follows ˆp f = n i=1 D fi n and ˆp l = n i=1 D li n (13) We calculate the following Z score, in order to test whether the frequency of the favourites best odds being offered by position taking bookmaker is random: ˆp f ˆp l ˆp(1 ˆp)(2/n), (14) where ˆp = n i=1 D fi + n 2n i=1 D li. (15) The null hypothesis is that price dispersion is random and therefore, there is no systematic tendency from position taking bookmakers setting outlying odds for favourites (i.e. pˆ f ˆp l = 0). 4.2.3 The Source of Returns by Bookmaker in the Arbitrage Portfolios. According to H3a, we expect that the leg responsible for the positive returns to the arbitrage portfolio will be placed with the position-taking bookmaker, on the favourite, whilst the longshot bet will be placed with negative or zero expectation at the book-balancing bookmaker. In order to test this hypothesis, we conduct a betting simulation, where a unit stake ($1) is placed on each bet that is selected from the linear program across the total sample of matches. For each type of bookmaker, we calculate for each match i and potential stake at offer j the bettors profit Z ij, as 6 Z i = S ij X ij + S i7 X i7 + 0.5(S i3 + S i10 ) + (S i4 + S i11 ) (16) j=1 13 +0.5(S i5 + S i12 + S i5 X i5 + S 12 X i12 ) + (S i6 + S i13 X i13 ) + S ij X ij 13 S ij j=8 j=1 where S ij is the amount staked on product j in match i. In the unit-stake simulation, S ij = 1 if there is a bet on offer j in match i, and zero otherwise. As a result, the average profit that the 25

Grant, Johnson, and Oikonomidis bettor achieves against each type of bookmaker, across the sample of n bets can be calculated as: µ = 1 n n 13 i=1 j=1 Z ij (17) If H3a is true, we should see that that µ is higher for the group of position-taking bookmakers than for the group of book-balancing bookmakers. It could be argued that the result of placing a unit stake across each bet is subject to high variance, since the average profit is highly influenced by the outcome of bets on longshots. Therefore, in order to ensure that latter bets are not leading to biased conclusions regarding the expected profit against each bookmaker, we replicate the simulation, where each stake S ij is determined by a proportional staking strategy, the Kelly Criterion (Kelly, 1956). Here we assume that for each type of bookmaker, all bets bear equal expected profit and, as a result, the application of the Kelly Criterion results in weighting each bet disproportionately to its odds. Hence, for each selected offer j of match i, instead of a unit stake, we bet with a stake size inversely proportional to the net odds offered by the bookmaker. S ij = 1 X ij 1 (18) As a result the average realized profit against each bookmaker across the sample of n bets is µ = n i=1 13 j=1 Z ijs ij n i=1 13 j=1 S ij (19) We recalculate the mean for each of the two types of bookmaker and compare them, in order to confirm that the conclusions drawn from the unit-stake simulations are not biased from abnormally positive or negative results on high-odds bets. 4.2.4 The Relative Efficacy of the Odds-Implied Bookmaker Probabilities. In order to test whether predictions based on book-balancers odds are more efficient (H3b) and unbiased (H3c) predictors of event outcomes compared to predictions based on position-takers odds, we compare the forecasting accuracy and the favourite longshot bias observed in predictions based on the odds of the two different types of bookmakers. To accomplish this we employ a conditional 26

Arb-Mirage: Are Inefficient Betting Markets an Illusion? logistic regression (with the probability of outcome o derived from the odds as the sole independent variable), where the outcome of each match is the dependent variable (i.e. home win, draw, or away win), which takes value 1 for the event that occurred and 0 for the events that did not occur. Hence, the probability that outcome o in match i occurs, is given by: P (Y io = 1) = e Z io ( 3 o=1 ez io ) (20) Z io is a function of the probability p io, as the latter is the probability of the event outcome implied by the odds for each outcome o of match i (the superscript s here implying the subjective probability based on the bookmaker and bettors combined assessment of the chance of this outcome), such that: Z io = b ln (p s io) (21) where p io can be calculated from the odds X io of outcome o in match i as p io = 1/X io (1 + ρ i ) (22) and ρ i is the bookmakers over-round. This can be calculated from the odds offered for all outcomes o of match i ρ i = 3 1/X io 1 (23) i=1 Hence, (20) can be written as: P (Y io = 1) = e (b ln (ps io )) 3 o=1 e(b ln (ps io )) = (p s io )b 3 o=1 (ps io )b (24) Positive favourite longshot bias indicates that the bookmaker odds underestimate the probability of the favoured event occurring. Therefore, if a bookmaker exhibits this bias, the actual winning probability of favourites, as implied by their observed frequency of success, is higher compared to that expected by the odds; whereas for the longshots it is lower. Thus, denoting as p v io the true 27

Grant, Johnson, and Oikonomidis probability (v denoting verifiable or objective) of outcome o in match i, we can infer the following: p s if > ps il pv if /pv if > pv il /ps il (25) where f denotes favourite, and l denotes longshot Subject to (24) p v if p s if > pv il p s il (ps if )b 3 o=1 (ps if )b > (p s il )b ( 3 o=1 (ps il )b ) (ps if )b 1 > (p s il )b 1 (26) When (p s if ) > (ps il ), (20) is only valid where b > 1. As a consequence, the odds of a given bookmaker underestimates favourites on average, only if b in (26) is significantly greater than 1 and higher values of b indicate higher degree of bias. Maximum likelihood is employed to estimate b and therefore, assess the degree of the bias in the odds offered by each bookmaker. To assess the accuracy of each bookmaker s predictions, we compare the values of McFadden s (1974) pseudo-r 2 statistic that each bookmaker s odds-implied probabilities achieve in the conditional logit model (a higher pseudo-r 2 implies a superior model fit and hence a greater degree of efficiency). 5 Results 5.1 Odds movement and liquidity at the betting exchange and a book-balancing bookmaker Table 4 presents summary statistics concerning the liquidity of the betting exchange Betfair and SBOBet, a major Asian (book-balancing) bookmaker, for 115 matches across the top six European leagues between 23/8/2013 and 1/9/2013. The liquidity, represented by the median stake size 21 available to the bettor, is measured at two different points in time: two days and two hours prior to kick-off. We consider the total amount that can be staked. Hence for Betfair, this is the sum of money available for all three Back and Lay offers (the depth), and for SBOBet, the sum of the maximum stake allowed on both sides. The associated round-trip transaction costs of placing 21 Bookmakers often set their limits in terms of payoffs (i.e. max. stake / (odds 1)), as they are primarily interested in limiting their exposure; the amount that they will have to pay in the event that the bet is successful. 28

Arb-Mirage: Are Inefficient Betting Markets an Illusion? a bet are also recorded. 22 At both points in time, there is higher liquidity and lower costs of trading at the book-balancing bookmaker. 23 The standard deviation of the maximum stake size indicates that that, while there are occasions when the betting exchange offers high liquidity, this is relatively rare in comparison to the bookmaker. Consequently, arbitrage opportunities observed with the book-balancing bookmaker should provide more meaningful economic significance than those requiring a position at the betting exchange. Table 4. Liquidity and transaction costs at a betting exchange and book-balancing bookmaker. This table reports the median staking limit and standard deviation of staking limits 2 days and 2 hours before match kickoff (K.O.) for a sample of 115 games in the largest six European leagues in the 2012-13 season (all matches played between 23/8/2013 and 1/9/2013). Provider Timing Number of Median Std. Dev. Median Matches Staking Limit Staking Limit Trans. Cost Betfair 2 days to K.O. 115 739 14,516 2.70% Betfair 2 hours to K.O. 115 11,592 69,089 2.10% SBOBet 2 days to K.O. 115 8,890 2,997 1.80% SBOBet 2 hours to K.O. 115 26,668 6,930 1.70% 5.2 Arbitrage Opportunities Across Bookmakers In this subsection we report on the odds of the bookmakers in our sample, and on the frequency and type of arbitrage opportunities, Table 5 reports the average closing odds correlation in our (main) sample of 2,132 matches across the six largest European football leagues in the 2011-12 season, for which prices were collected simultaneously. The odds offered by diverse market operators are highly correlated, demonstrating that on average the bookmakers offers are reasonably well-aligned. The highest level of correlation is observed amongst book-balancing bookmakers. Despite the fact that these bookmakers move their odds more frequently 24 (adjusting for individual high stakes), 22 For Betfair, the transaction cost for each offer is calculated as the over-round (sum of the inverse odds) considering the (net of Betfair s 2% commission on profit) volume-weighted average of odds for Backing (betting in favour of that team) and the volume-weighted average of odds for laying (betting against that team). For SBOBet, the transaction cost for each game corresponds to the over-round considering the two offers for them main handicap line (e.g. +0.5 handicap team A and -0.5 handicap team B, on A vs B). 23 Once a trader takes all offers on Betfair s screen, they will have to wait for new offers to be provided by the public, even if they are willing to trade at lower odds. SBOBet and other book-balancing bookmakers will immediately place a new offer with equal or slightly decreased odds, sustaining the provision of liquidity. As a result, the figures in Table 4 are likely to understate the liquidity differences across the two markets. Moreover, it should be considered that there are a number of book-balancing bookmakers who accept stakes of the same magnitude as SBOBet. Other than Betfair, there is no betting exchange where significant volumes can be traded. 24 The frequency of odds movements is a key characteristic of book-balancing bookmakers, in order to retain low inventory positions and offer low transaction costs. 29

Grant, Johnson, and Oikonomidis such adjustments seem to happen in parallel across the set of book balancing bookmakers group, or at least do not appear to impact the closing odds greatly. Among position-takers, Bet365 seems to be the bookmaker most aligned with the book-balancers (possibly due to the fact that it is the only bookmaker in the group for which online betting is its sole focus), whereas Ladbrokes and especially Stan James are the operators showing the least correlation with the book-balancing market. Table 5. Average closing odds correlation between bookmakers in sample of 2,132 games in 2012-13 European football league matches. Panel A reports the correlation matrix of closing odds across all bookmakers, Panel B reports the average correlation between and within groups of book-balancing and position-taking bookmakers. Panel A: Correlation in Odds Between Bookmakers Position Takers Book Balancers Bookmaker Ladbrokes William Hill Bet 365 Stan James SBO Bet 188Bet Pinnacle Ladbrokes 1.0000 Position William Hill 0.9927 1.0000 Takers Bet 365 0.9907 0.9950 1.0000 Stan James 0.9928 0.9917 0.9891 1.0000 Book SBOBet 0.9884 0.9925 0.9947 0.9849 1.0000 Balancers 188Bet 0.9890 0.9942 0.9970 0.9867 0.9971 1.0000 Pinnacle 0.9890 0.9944 0.9971 0.9868 0.9971 0.9987 1.0000 Panel B: Average Correlation Within and Across Groups Group 1 Group 2 Average Position Takers Book Balancers 0.9913 Position Takers Position Takers 0.9920 Book Balancers Book Balancers 0.9976 The optimization process presented in (4) to (9) reveals the existence of 545 arbitrage opportunities across the 2,132 matches in our sample with simultaneous prices, or in 25.6% of matches. The distribution of these opportunities across the different leagues is shown in Table 6 and the frequency with which each bookmaker s odds feature in the optimized portfolio are shown in Table 7. The arbitrage opportunities are well spread out across the leagues, although the relatively low frequency in Holland returns a χ 2 (5) test of independence with a p-value of 0.048. As Holland is the least popular of the six major leagues, there is less competition among bookmakers to supply competitive odds, and as such, transaction costs are higher. Hence it is not surprising that the fewest number of arbitrage opportunities occur in the Dutch Eredivisie. The linear program (4) to (9) can result in the odds of a diverse number of bookmakers featuring in each potential arbitrage opportunity (ranging from 2 to 6 in our sample), in order to achieve the maximum risk-free profit. In cases where multiple bookmakers post equal maximum odds for the 30

Arb-Mirage: Are Inefficient Betting Markets an Illusion? Table 6. Number of Matches with Arbitrage Opportunities by League. League Number of Matches with Frequency Matches in Sample Arbitrage Opportunity England 90 16.50% 380 Spain 109 20.00% 380 Italy 101 18.50% 380 Germany 94 17.20% 306 France 94 17.20% 380 Holland 57 10.50% 306 TOTAL 545 100.00% 2,132 same market, we attribute each of the bookmakers as having supplied the arbitrage opportunity. It is clear from Table 7 that some bookmakers are more likely than others to be involved in the generation of a theoretically risk-free portfolio. 25 The frequency of the appearance of positiontaking bookmakers is likely to be related each operator s policy on promotional odds. A test of independence rejects that bookmakers appear with equal frequency in arbitrage portfolios (χ 2 (6) = 615.4, p-value = 0.000). Table 7. Number of times a bookmaker s odds feature in a potential arbitrage portfolio selected by the optimisation programme for the 2,132 league matches played in the major European leagues in the season 2012-13. For each bookmaker we report the relative frequency with which their odds appeared in the X max vector, and featured in the optimal potential arbitrage portfolio identified for a single game. Group Position- Takers Book-Balancers Bookmaker Times odds featured in arbitrage portfolio Relative Frequency Ladbrokes 237 13.27% William Hill 59 3.30% Bet 365 65 3.64% Stan James 452 25.31% SBOBet 248 13.89% 188Bet 262 14.67% Pinnacle 463 25.92% TOTAL 1786 100% Consistent with the behaviour suggested by the inter-bookmaker correlation statistics, Bet365 and William Hill appear to be more aligned in their pricing strategy with the book balancing 25 Removing Stan James from the sample causes the instances of potential arbitrage opportunities to drop to 287, which is indicative of the influence of a bookmaker which applies a policy of offering outlying odds, on the creation of arbitrage instances. 31

Grant, Johnson, and Oikonomidis bookmakers, and hence their odds appear less frequently in arbitrage portfolios. On the other hand, all book-balancing bookmakers are often part of an arbitrage portfolio, with Stan James or Ladbrokes often being on the opposite side (i.e. offering the opportunity to hedge the stake placed against the book-balancers). Amongst the book-balancing bookmakers, Pinnacle is selected most frequently by the optimization program to be part of an arbitrage portfolio. This may arise because this is the only book balancing bookmaker which exhibits the same low over-round on 1 2 markets as on the Asian Handicap offers. 26 As a result, Pinnacle often offers the highest odds on a draw, which is frequently a useful bet in terms of equalizing payoffs across all outcomes. 27 If a bettor could have maintained access to all 7 bookmakers, without suffering restrictions in her betting size, the fully hedged strategy would have returned an impressive 7.56 times the initial bankroll across the season (assuming no reinvestment). This corresponds to an average risk-free profit of 1.38% per match. In order to realistically measure the anticipated yield of such a strategy, one should account for complications relating to the occurrence of overlapping times of matches, and the application of staking limits from the bookmakers for the various offers, which are likely to restrict profitability for sizeable bankrolls. Taken together, the results discussed above show that sufficient price dispersion exists in the market for bettors to create a seemingly risk-free portfolio of bets that would guarantee to them profits for about 25% of football games played, if we assume that they could successfully implement this strategy. These results serve to support Hypothesis 1; there exist instances where the price dispersion in the betting market is adequate to generate theoretically risk-free opportunities for bettors to profit by simultaneously betting with different bookmakers on alternative outcomes related to the same event. 5.3 Arbitrage Opportunities by Type of Bookmaker We determine for each apparent arbitrage opportunity the source of the odds which make up the arbitrage portfolio of bets (i.e. from book balancing or position taking bookmakers). In particular, we look at the instances where the odds offered by book-balancing or position-taking bookmakers 26 Book-balancers maintain an over-round of about 2% in Asian Handicap markets, but their over-round is nearer 5% 6% for the 1 2 market. On the other hand, Pinnacle s over-round is about 2% in the 1 2 market. 27 Mainly when a positive handicap (i.e. either +0.25 or +0.5) is not selected, the optimisation indicates a stake should be placed on the draw so that there is no negative exposure on the draw outcome. 32

Arb-Mirage: Are Inefficient Betting Markets an Illusion? on the favourite or the longshot feature in the arbitrage opportunities identified in our sample of matches. These results are displayed in Table 8. In 84% of arbitrage opportunities, one or more position-takers will offer the highest odds on the favourite (longshot) while one or more book-balancers will offer the highest odds on the longshot (favourite). Based on this frequency of arbitrage opportunities being created by bookmakers of different types (book-balancers and position-takers), the chance that this phenomenon is random is very low (Z-statistic = 16.12, p- value=0.000). This finding supports Hypothesis 2; most apparent arbitrage opportunities involve bets placed with different types of bookmakers (book-balancers and position-takers). Table 8. Constituent bets in arbitrage portfolios by type of bookmaker and odds. The first (second) column shows the type of bookmaker for which the optimal arbitrage portfolios contain bets on favourites (longshots). The favourite is identified as the team with the lower odds on the 1X2 betting market, bets on products 1-6 in Table 2 are considered favourite bets if the home team has lower odds; bets on products 8-13 in Table 2 are considered favourite bets if the away team has lower odds. All other bets (excluding draw bets) are considered longshot bets. The number and proportion for which arbitrage portfolios are constructed, using bets from each type of bookmaker, are presented in column 3. Columns 4 and 5 report similar results to column 3, with the strength of the favourite increasing to $2.00 and $1.70 per dollar bet. Column 6 repeats the results from column 3 with the exclusion of the outlying position-taking bookmaker Stan James. Best Offer Best Offer Num. matches Num. matches Num. matches Num. matches Favourite by: Longshot by: (full sample) (Fav < 2.00) (Fav < 1.70) (w/o Stan James) prop prop prop prop Position Position 41 27 6 18 Taker Taker 0.075 0.071 0.073 0.061 Position Book 277 183 57 152 Taker Balancer 0.508 0.480 0.695 0.514 Book Position 181 142 14 78 Balancer Taker 0.332 0.373 0.171 0.264 Book Book 46 29 5 48 Balancer Balancer 0.084 0.076 0.061 0.162 Total 545 381 82 296 1.000 1.000 1.000 1.000 Given the considerable differences in how the two types of bookmakers operate, this finding is related to that of Franck et al. (2013), who found arbitrage opportunities occurred significantly more often across market-types (between bookmaker and exchange) than within bookmakers. Our results suggest that the intensive mobility of the book-balancers odds occasionally drives them away from the position-takers odds. The response to price movements by position-taking bookmakers depends on their marketing-related objectives. As a result, opportunities for seemingly risk-free 33

Grant, Johnson, and Oikonomidis profit arise by combining the odds offered by position-takers and book-balancers on the same event. On the other hand, intra-market competition is likely to lead to high price-coordination within each group (i.e. position- takers or book-balancers), restricting the likelihood of high price-dispersion between two bookmakers in the same group. Consequently, instances of arbitrage opportunities between bookmakers of the same group rarely arise. The results displayed in Table 8 also show that position takers are significantly more likely to offer above market odds for the favourite rather than for the longshot. On 58% of occasions, a position-taking bookmaker offered the highest odds for the favourite, compared with 44.4% offering the highest odds on the longshot. Such a difference is unlikely to be random (Z-statistic = 5.86, p-value = 0.000). This tendency is more pronounced on stronger favourites (i.e. the more heavily favoured teams). Since position-taking bookmakers attract higher volumes on the favourites than on the longshots (Levitt, 2004), our finding is consistent with the view that position-taking bookmakers 28 are inclined to inflate odds for popular bets in order to attract customers. As a result, arbitrage opportunities most commonly emerge where a position-taker offers the highest odds for the favourite and a book-balancer offers the highest odds in the market for the longshot. 29 These results are in line with Hypothesis 2. 5.4 Winners and Losers from Apparent Arbitrage Opportunities In order identify the type of bookmakers which would lose against potential arbitrageurs, should the identified risk-free opportunities be exploitable, we employ the simulation described in (16). Placing $1 on all 813 outcomes for which the odds posted by position taking bookmakers form part of the fully-hedged portfolio, yields an average profit of $0.16 per bet. Adopting the same strategy of backing all outcomes where the book-balancers odds feature in the optimal fully-hedged portfolio results in a loss of $0.024 per bet. 30 The profit obtained on the bets placed at the position- 28 Such studies do not distinguish between diverse types of operators, but they assume a type of bookmaker consistent with our position-takers definition. 29 In general, position-takers, do not offer higher odds for favourites on average compared to book-balancers due to their higher over-round). However, their odds on favourites are closer to those offered by book-balancers than the odds they offer on longshots. This finding suggests that they probably do not distribute their over-round proportionally. 30 It has to be clarified that in this case $1 is bet on each offer that falls part of the portfolio, no matter what the fraction of capital allocated from the optimization (4). Therefore, the results of this simulation are not comparable to the results of the fully hedged strategy. By way of example, the fully hedged strategy may assign 90% of the capital to bet A and 10% to bet B and hence, we would bet $0.90 and $0.10 on these products, respectively. However, in the unit-stake simulation $1 is staked on bet A and $1 on bet B, since the objective is to identify how the profit is distributed across the two types of bookmakers, rather than to create a hedged position. 34

Arb-Mirage: Are Inefficient Betting Markets an Illusion? takers odds in these cases is significantly higher than the profit (in fact a loss) obtained on the bets placed at the position takers odds (t-statistic = 3.37, p-value = 0.000). The high return against the position-takers though is exaggerated by the fact that a number of high paying longshots won in this particular sample. 31 Adjusting the strategy as described by equation (19) leads to an average profit per bet of $0.04 per $1 stake against position taking bookmakers and an overage loss of $0.047 per $1 stake against book-balancers. These returns remain significantly different (t-statistic = 2.60, p-value = 0.005). This result supports Hypothesis 3a; position-takers suffer losses on average when an apparent arbitrage opportunity exists. Interestingly, the loss incurred by the book-balancing bookmakers on these bets is close to their over-round. 32 The component of arbitrage portfolios placed through book-balancers generates returns for a bettor equal their expected loss had they placed a random bet with these bookmakers. In other words, the fact that another bookmaker offers sufficiently different odds to generate an apparent arbitrage opportunity does not change the expected value of the bets that they receive. Hence, this result effectively justifies Pinnacle s statement that the motive for a bet (e.g. intention to arbitrage) should be irrelevant to a book-balancer. 33 From the bettor s perspective it seems that a higher return is expected by placing bets against outlying odds of position-takers, rather than by hedging such positions against book-balancers, since in the latter case the average profit drops to $0.013 per $1 bet. Consequently, there is evidence from these simulations to support Hypothesis 3a; the expected loss from an apparent arbitrage opportunity is likely to be suffered by the position-taking bookmakers. This is consistent with the findings of Franck et al. (2013). 5.5 The Efficiency of Bookmakers Odds by Type Table 9 presents the results of estimating separate conditional logistic regression models (as described in (20) and (24)) based on the 1 2 odds offered on football matches across the six leading 31 55 out of 813 $1 bets that were placed at gross odds greater than $5 had an extremely high profit of $1.30. As a result, this small number of lucky bets account for $71.60 out of the $130 won in total by this strategy. Hence it is important to ensure that they do not bias the conclusions. This is achieved by applying the weighting implied by equation (19). 32 The over-round of such bookmakers is about 2% for Asian Handicap offers and 5% 6% for 1 2 offers, excluding Pinnacle, which employs an over-round of around 2% in the 1 2 market. 33 Pinnacle s statement is: [A]ll bookmakers shouldn t care about the motivation for placing a bet, but should simply look to balance the bet volume. Source: http://www.pinnaclesports.com/betting-promotions/ arbitrage-friendly. 35

Grant, Johnson, and Oikonomidis European leagues by position-taking and book-balancing bookmakers, respectively, for our extended sample of 6,396 matches over the three seasons from 2009-10 to 2011-12. The forecasting accuracy of odds offered by book-balancing bookmakers is higher on average compared to that of position takers odds, as represented by the higher average McFadden s pseudo-r 2. Whilst the increase in pseudo-r 2 appears small, it is likely to be economically significant (e.g. Benter, 1994). This result is in line with Hypothesis 3b and is consistent with the evidence provided by Franck et al. (2013) and (Smith, Paton and Vaughan Williams, 2006, 2009) that demand-driven (as opposed to traditional position-taking bookmaker) markets are more efficient predictors of event outcomes. The values of the coefficients on the conditional logistic regressions indicate that the favouritelongshot bias is more pronounced for position-takers than for book-balancers, confirming Hypothesis 3c. The notable exception is for the odds offered by the book balancer SBOBet, whose over-round for the 1 2 market is the highest amongst book-balancing bookmakers. The conditional logistic regression based on Pinnacle s odds (whose over-round in the 1 2 market is as low as it is for Asian Handicaps) has a coefficient very close to 1, suggesting no favourite-longshot bias. This variation of the bias is in line with the Vaughan Williams (1999) proposition that the level of transaction costs affect the degree of favourite-longshot bias. The extent of the bias is likely to indicate that book-balancing bookmakers (excluding Pinnacle) aim to direct the demand for longshots to Asian Handicap markets, in order to facilitate the balancing of their book in that market. 6 Discussion In this paper, we analysed a unique data set of 1 2 and Asian Handicap odds for football games played in major European leagues, offered simultaneously by the several large bookmakers. We collected match odds at times close to the kick-off, when markets are most liquid. Employing a linear programming methodology, we identified the best combination for each of 545 games where a fully hedged profitable investment appears to be possible. Notwithstanding challenges related to its implementation, such a strategy could, in theory, guarantee a profit of 1.3% per game on average. To some extent, our findings confirm those of Franck et al. (2013); arbitrage opportunities mainly exist across, rather than market structures. However, with our unique data set, we can be more certain that the disparate odds required to form an arbitrage portfolio were concurrently 36

Arb-Mirage: Are Inefficient Betting Markets an Illusion? Table 9. This table reports the results of conditional logit modelling (using equations (20) and (24)) based on the odds offered by six bookmakers on all 6,396 matches in the English, Spanish, Italian, German, French and Dutch leagues over the three seasons 2009-10, 2010-11 and 2011-12 for all match outcomes (Home win, Draw, Away win; 19,188 total observations of odds per bookmaker). Bookmakers are classified as either position-takers or book-balancers. The third and fourth columns of the table report the estimated coefficient of the conditional logit model and its standard error, respectively. The fifth column reports the p-value of a Z-test to determine whether the true value of the coefficient in column three is equal to 1. The sixth column reports the result of test whether the coefficient is significantly greater than 1 at the 10%, 5%, and 1% levels with the signs ( ), ( ), and ( ), respectively. The final column reports the McFadden Pseudo-R 2 of the conditional logit model. Group Bookmaker Coefficient Std. Error Position Takers Book Balancers Prob. (Coeff. = 1) Sig. Pseudo-R 2 Ladbrokes 1.0743 0.0309 0.0081 (***) 0.1085 William Hill 1.0605 0.0304 0.0232 (**) 0.1101 Bet365 1.0560 0.0302 0.0318 (**) 0.1106 SBOBet 1.0784 0.0311 0.0059 (***) 0.1106 188Bet 1.0413 0.0299 0.0831 (*) 0.1113 Pinnacle 1.0081 0.0289 0.3900 0.1114 available, and sufficiently liquid to be exploited. Moreover, all of our odds were obtained from bookmakers, who have been considered mainly in homogenous terms in prior studies. Our study suggests a dichotomy of bookmakers into position-takers and book-balancers, who mainly differ in terms of clientele. Position-takers restrict service to sophisticated bettors and hold physical-world locations, mainly in Europe where 1 2 betting is prevalent. Book-balancers do not restrict service to informed customers, and are prominent operators in Asia. Due to their presence in this study, we use Asian Handicaps as part of the arbitrage portfolio. We build upon the work Marshall (2009), who explored realistic arbitrage opportunities in betting markets, and found that arbitrage opportunities between bookmakers are not infrequent. We suggest that this finding, which appears to contradict that of other studies (e.g., Dixon and Pope, 2004; Deschamps and Gergaud, 2007; Luckner and Weinhardt, 2008; Deschamps, 2008; Vlastakis et al., 2009; Franck, et al., 2013) arises from the choice of bookmakers under investigation. Studies that consider position-taking, or book-balancing bookmakers only are expected to identify a significantly lower frequency of potential arbitrage instances compared to studies that consider both types of bookmakers together. Investigating the bookmaker-specific attributes of the parties involved in the apparent arbitrage 37

Grant, Johnson, and Oikonomidis opportunities leads to the conclusion that such opportunities are likely to be created by positiontakers inefficient pricing. This pricing policy maybe intentional, in order to publish odds that may attract customers, or the result of their prices lagging behind book-balancers (due to the pace at which book-balancers odds are informatively updated, driven by the flow of smart money ). 34. Given the public availability of odds, it seems fair to assume that if the sole objective of the price setting strategies of position-taking bookmakers were the efficient calibration of event outcomes they would fully align their odds with those of book-balancers. The fact they do not do so indicates that odds arise as a result of strategic behaviour. Pricing strategies include offering higher odds for popular bets compared to their competitors or keeping stable odds in order to avoid upsetting casual bettors by continuous changes. 35 Combining the facts that position-takers hold imbalanced books, and are aware of their setting of inefficient prices with high transaction costs, we conclude that the restriction of sophisticated bettors is part of their strategy. The setting of promotional odds partially assists the position-taking bookmakers in identifying informed clients, who tend to place bets only at prices with negative expectation for the bookmaker. In addition, significant anecdotal evidence exists that such bookmakers operate discriminating behaviour, against longterm winning customers (Franck et al., 2013). Consequently, we would argue that the majority of apparent arbitrage opportunities observed in fixed odds betting markets are very likely to be a mirage. There are broad implications for a lack of calibration in bookmakers price-setting objectives. Efficient pricing in betting markets, as in wider financial markets, implies prices that accurately reflect the fundamental value of assets. However, it seems that in the most popular forms of betting, bookmakers may consider alternative objectives when setting its odds, which may lead to (even intentional) mispricing. Such mispricing is unlikely to be systematically exploitable, due to restrictions imposed on potentially skilled bettors. In that sense, the original concept of the EMH, according to which market prices are expected to converge to the fundamental values, subject to the activity of informed traders, may be radically distorted. The field experiment presented by betting markets suggests that separating uninformed traders 34 Pinnacle state This limiting of arbitrage players is a reflection of a bookmaker s short-comings, such as posting bad odds, or an inability to move odds fast enough to avoid being the focus of arbitrage players. Source: https: //www.pinnaclesports.com/betting-promotions/arbitrage-friendly. 35 Frequent odds moves often lead to bets failing to execute. Moreover, bookmakers with physical locations incur larger costs when changing prices relative to online-only firms. 38

Arb-Mirage: Are Inefficient Betting Markets an Illusion? from the aggregate market is costly. The client profile management conducted by position-taking bookmakers to select only suitably uninformed counterparties to trade with places a higher cost on bettors, in terms of transaction prices and search frictions for future trades. The loss of reputation to such bookmakers around client exclusion is likely to lead to a broad shift in market share towards the less restrictive book-balancing bookmakers or betting exchanges in the long run. Dealers in wider financial markets should consider the implications of short-term profit maximising strategies against the evolution of competing market mechanisms in deciding appropriate counterparties with whom to trade. References Adams, Brian R., Frank W. Rusco, and W. David Walls (2002) Professional Bettors, Odds Arbitrage Competition, and Betting Market Equilibrium. Singapore Economic Review 47, 111 127 Akbas, Ferhat, Will J. Armstrong, Sorin Sorescu, and Avanidhar Subrahmanyam (2014) Capital Market Efficiency and Arbitrage Efficacy. Journal of Financial and Quantiative Analysis Forthcoming, xx xx Benter, William (1994) Computer-Based Handicapping and Wagering Systems: A Report. In Efficiency of Racetrack Betting Markets, ed. Victor S. Y. Lo, Donald B. Hausch, and Willam T. Ziemba Academic Press New York pp. 183 192 Bernard, Victor L., and Jacob K. Thomas (1989) Post-Earnings-Announcement Drift: Delayed Price Response or Risk Premium? Journal of Accounting Research 27, 1 36 Brunnermeier, Markus K. (2009) Deciphering the Liquidity and Credit Crunch 2007-08. Journal of Economic Perspectives 23, 77 100 Croxson, Karen, and J. James Reade (2011) Exchange vs. Dealers: A High-Frequency Analysis of In-Play Betting Prices. Department of Economics, Discussion Paper 11-19, University of Birmingham (2014) Information and Efficiency: Goal Arrival in Soccer Betting. The Economic Journal 124, 62 91 39

Grant, Johnson, and Oikonomidis Dantzig, George B. (1951) Maximization of a Linear Function of Variables Subject to Linear Inequalities. In Activity Analysis of Production and Allocation, ed. T.C. Koopmans John Wiley and Sons New York pp. 339 347 Deschamps, Bruno (2008) Betting Markets Efficiency: Evidence From European Football. Journal of Gambling Business and Economics 2, 66 76 Deschamps, Bruno, and Olivier Gergaud (2007) Efficiency in Betting Markets: Evidence from English Football. Journal of Prediction Markets 1, 61 73 Direr, Alexis (2013) Are Betting Markets Efficient? Evidence from European Football Championships. Applied Economics 45, 343 356 Dixon, Mark J., and Peter F. Pope (2004) The Value of Statistical Forecasts in the UK Association Football Betting Market. International Journal of Forecasting 20, 697 711 Duffie, Darrell (2012) Dark Markets: Asset Pricing and Information Transmission in Over-The- Counter Markets (Princeton, NJ: Princeton University Press) Edelman, David C., and Nigel R. O Brian (2004) Tote Arbitrage and Lock Opportunities in Racetrack Betting. European Journal of Finance 10, 370 378 Fama, Eugene F. (1970) Efficient Capital Markets: A Review and Theory and Empirical Work. Journal of Finance 25, 383 417 Fingleton, John, and Patrick Waldron (1999) Optimal Determination of Bookmakers Betting Odds: Theory and Tests. Department of Economics, Discussion Paper 96/9, Trinity College Dublin Forrest, David (2012) The Threat to Football from Betting-Related Corruption. International Journal of Sports Finance 7, 99 116 Forrest, David, and Robert Simmons (2008) Sentiment in the Betting Market on Spanish Football. Applied Economics 40, 119 126 40

Arb-Mirage: Are Inefficient Betting Markets an Illusion? Franck, Egon, Erwin Verbeek, and Stephan Nüesch (2010) Prediction Accuracy of Different Market Structures - Bookmakers Versus a Betting Exchange. International Journal of Forecasting 26, 448 459 (2011) Sentimental Preferences and the Organizational Regime of Betting Markets. Southern Economic Journal 78, 502 518 (2013) Inter-market Arbitrage in Betting. Economica 80, 300 325 Friedman, Milton (1953) Essays in Positive Economics (Chicago: University of Chicago Press) Froot, Kenneth A., and Jeremy C. Stein (1998) Risk Management, Capital Budgeting, and Capital Structure Policy for Financial Institutions: An Integrated Approach. Journal of Financial Economics 47, 55 82 Gramm, Marshall, and Douglas H. Owens (2005) Determinants of Betting Market Efficiency. Applied Economics Letters 12, 181 185 Gromb, Denis, and Dimitri Vayanos (2010) Limits of Arbitrage: The State of the Theory. NBER Working Paper No. 15821 Hausch, Donald B., and William T. Ziemba (1990) Arbitrage Strategies for Cross-Track Betting on Major Horseraces. Journal of Business 63, 61 78 Ho, Thomas S. Y., and Hans R. Stoll (1983) The Dynamics of Dealer Markets Under Competition. Journal of Finance 38, 1053 1074 Hodges, Stewart, Hao Lin, and Lan Liu (2013) Fixed Odds Bookmaking with Stochastic Betting Demands. European Financial Management 19, 399 417 Humphreys, Brad R. (2011) The Financial Consequences of Unbalanced Betting on NFL Games. International Journal of Sports Finance 6, 60 71 Kelly, John L. (1956) A New Interpretation of Information Rate. Bell Systems Technical Journal 35, 917 926 Kuypers, Tim (2000) Information and Efficiency: An Empirical Study of a Fixed Odds Market. Applied Economics 32, 1353 1363 41

Grant, Johnson, and Oikonomidis Lane, Daniel, and William T. Ziemba (2004) Jai Alai Arbitrage Strategies. European Journal of Finance 10, 353 369 Levitt, Steven D. (2004) Why are Gambling Markets Organized So Differently from Financial Markets? The Economic Journal 114, 223 246 Levitt, Steven D., and John A. List (2007) What Do Laboratory Experiments Measuring Social Preferences Reveal About the Real World? Journal of Economic Perspectives 21, 153 174 Luckner, Stefan, and Christof Weinhardt (2008) Arbitrage Opportunities and Market-Making Traders in Prediction Markets. In 10th IEEE Conference on E-Commerce Technology and the Fifth IEEE Conference on Enterprise Computing, E-Commerce and E-Services IEEE Washington, D.C. pp. 53 59 Madhavan, Ananth (2000) Market Microstructure: A Survey. Journal of Financial Markets 3, 205 258 Magee, Sean (1990) The Channel Four Book of Racing (London, U.K.: Sidgwick and Johnson) Marshall, Ben R. (2009) How Quickly is Temporary Market Inefficiency Removed? Quarterly Review of Economics and Finance 49, 917 930 McFadden, Daniel L. (1974) Conditional Logit Analysis of Qualitative Choice Behavior. In Frontiers in Econometrics, ed. P. Zarembka Academic Press New York pp. 105 142 McLean, R. David, and Jeffrey Pontiff (2014) Does Academic Research Destroy Stock Return Predictability? mimeo, University of Alberta and Boston College Naik, Narayan Y., and Pradeep K. Yadav (2003) Do Dealer Firms Manage Inventory on a Stockby-Stock or a Portfolio Basis? Journal of Financial Economics 69, 325 353 Oikonomidis, Anastasios, and Johnnie Johnson (2011) Who Can Beat the Odds? The Case of Football Betting Reviewed. In Prediction Markets: Theory and Applications, ed. Leighton Vaughan Williams Routledge Milton Park, U.K. pp. 204 220 Paton, David, and Leighton Vaughan Williams (2005) Forecasting Outcomes in Spread Betting Markets: Can Bettors Use Quarbs to Beat the Book? Journal of Forecasting 24, 139 154 42

Arb-Mirage: Are Inefficient Betting Markets an Illusion? Paul, Rodney J., and Andrew P. Weinbach (2007) Does Sportsbook.com Set Pointspreads to Maximize Profits? Tests of the Levitt Model of Sportsbook Behavior. Journal of Prediction Markets 1, 209 218 (2008) Price Setting in the NBA Gambling Market: Tests of the Levitt Model of Sportsbook Behavior. International Journal of Sports Finance 3, 137 145 Pontiff, Jeffrey (1996) Costly Arbitrage: Evidence from Closed-End Funds. The Quarterly Journal of Economics 111, 1135 1151 Pope, Peter F., and David A. Peel (1989) Information, Prices, and Efficiency in a Fixed-Odds Betting Market. Economica 56, 323 341 Rouwenhorst, K. Geert (1998) International Momentum Strategies. Journal of Finance 53, 267 284 Sauer, Raymond D. (1998) The Economics of Wagering Markets. Journal of Economic Literature 36, 2021 2064 Smith, Michael A., David Paton, and Leighton Vaughan Williams (2006) Market Efficiency in Person-to-Person Betting. Economica 73, 673 689 (2009) Do Bookmakers Possess Superior Skills to Bettors in Predicting Outcomes? Journal of Economic Behavior and Organization 71, 539 549 Spann, Martin, and Bernd Skiera (2009) Sports Forecasting: A Comparison of the Forecast Accuracy of Prediction Markets, Betting Odds and Tipsters. Journal of Forecasting 28, 55 72 Thaler, Richard H., and William T. Ziemba (1988) Anomalies: Parimutuel Betting Markets: Racetracks and Lotteries. Journal of Economic Perspectives 2, 161 174 Vaughan Williams, Leighton (1999) Information Efficiency in Betting Markets: A Survey. Bulletin of Economic Research 51, 1 39 Vaughan Williams, Leighton, ed. (2005) Information Efficiency in Financial and Betting Markets Cambridge University Press New York 43

Grant, Johnson, and Oikonomidis Vlastakis, Nikolaos, George Dotsis, and Raphael N. Markellos (2009) How Efficient is the European Football Betting Market? Evidence from Arbitrage and Trading Strategies. Journal of Forecasting 28, 426 444 Willis, Kenneth E. (1964) Optimum No-Risk Strategy for Win-Place Pari-Mutuel Betting. Management Science 10, 574 577 Woodland, Bill M., and Linda M. Woodland (1991) The Effects of Risk-Aversion on Wagering: Point Spread Versus Odds. Journal of Political Economy 99, 638 653 Zhu, Haoxiang (2014) Do Dark Pools Harm Price Discovery? Review of Financial Studies 27, 747 789 44