Tag Archives: pari

Equité et discrimination en assurance

Mercredi 18 mai, j’interviendrai au séminaire de la chaire PARI, à Paris, dans les locaux de la CCR, pour parler d’équité et de discrimination en assurance… Les slides sont en ligne.

En poursuivant la discussion abordée dans “L’équité de l’apprentissage machine en assurance” (co-écrit avec Laurence Barry), nous présentons un aperçu des problèmes auxquels les actuaires sont confrontés lorsqu’ils traitent de la discrimination. Depuis le début de leur histoire, les assureurs sont connus pour utiliser des données pour classifier et tarifer les risques. En tant que tels, ils ont été confrontés très tôt au problème de l’équité et de la discrimination associées aux données. Cette question devient de plus en plus importante avec l’accès à des données plus granulaires et comportementales, et évolue pour refléter les technologies et les préoccupations sociétales actuelles. En examinant les débats antérieurs sur la discrimination, nous montrons que certains préjugés algorithmiques sont une version renouvelée de préjugés plus anciens, tandis que d’autres semblent inverser l’ordre précédent. Paradoxalement, alors que la pratique de l’assurance n’a pas profondément changé et que la plupart de ces biais ne sont pas nouveaux, l’ère de l’apprentissage automatique ébranle encore profondément la conception de l’équité en matière d’assurance. (Cette présentation s’appuiera également sur un rapport qui sera bientôt publié par l’Institut Louis Bachelier).

Insurance against Natural Catastrophes: Balancing Actuarial Fairness and Social Solidarity

Our research paper, Insurance against Natural Catastrophes: Balancing Actuarial Fairness and Social Solidarity, with Molly James and Laurence Barry, is now available.

Natural disasters offer a special case for the study of private and public insurance mix. Indeed, the experience accumulated over the past decades has made it possible to transform poorly known hazards, long considered uninsurable, into risks that can be assessed with some precision. They exemplify however the limits of the risk-based premiums method, as it might imply unaffordability for some. The French scheme reflects such ideas and offers a wide coverage for moderate premiums to all, but is shaken by climate change: we show that some wealthier areas, that were not perceived as “at risk” in the past, have become exposed to submersion risk in the future. This singularly makes some well-off properties the potential main beneficiaries of a scheme that was historically thought to protect the worst-off. Acknowledging that some segmentation might become desirable, we examine several models for flood risk and the disparity in premiums they entail.

Intervention au séminaire TransNum de Sciences Po

Vendredi, j’interviendrai avec les collègues de la chaire PARI au séminaire TransNum à Sciences Po, à Paris, sur le thème Des risques individualisables ? Enjeux et limites pour le secteur assurantiel. 

Depuis la fin du XVIIIe siècle, l’industrie assurantielle repose sur un principe de mutualisation : tous (ou en tout cas un grand nombre) cotisent pour se couvrir contre un risque qui ne frappera que quelques-uns. La différence entre les sommes cotisées et celles qui sont versées rémunèrent les assureurs, dont le modèle économique repose sur la mobilisation d’outils mathématiques – les probabilités – dont le développement accompagne celui de cette industrie. L’arrivée des données massives est-elle susceptible de remettre en cause ce modèle ? La discussion sera menée d’un triple point de vue : historique, épistémologique et économique.

Les slides de mon intervention sont en ligne.

From betting to “prediction market”

This is the second part of a series on sports betting

Sports betting has long fascinated economists and statisticians. Griffith (1949) showed early on that horse race bettors put too much money on horses that have little chance of winning, and too little on those that have the best chance of winning. This tendency to underbid on the most likely event has been obaserved in all sports betting, where the “most likely event” is calculated on the basis of recent statistics. And it can be explained in a fundamental way by the mechanics of mutual betting: the bettor opposes his beliefs to those of the crowd, because the various bets will determine the odds.

Predictions, before surveys

Today, in the months leading up to each election, we find ourselves drowned under the polls, conducted every day (and commented on several times a day, as if estimation noise was worth exegesis). As Frédéric Dabi (Deputy Director General of Ifop) pointed out in a debate organised by Risques magazine in 2017, “surveys are an indication of the electoral balance of power, not a prediction”, but it is nevertheless often in the idea of having a prediction that they are used.

But if we go back in time, Rhode & Strumpf (2008) reminds us that other techniques were used, before the idea of surveys became necessary, in particular betting. In 1549, Matteo Dandolo (Ambassador of Veneto) noted that “it is therefore more than clear that the traders are very well informed of the state of the election, and that the employees of the cardinals in conclave (i conclavisti) participate with them in betting, which therefore leads to several tens of thousands of crowns changing hands” as Baumgartner (2003) tells us. Closer to home, betting markets during the elections were popular in the United States until the Second World War. Rhode & Strumpf (2008) suggests several reasons for the loss of interest in the second half of the 20th century: improvements in sampling techniques… and the legalization of horse betting. But online betting sites have revived interest in betting, whatever it may be. Because the sites we mentioned in a previous article are often not limited to sports betting, but also allow betting on a magnitude earthquake, an Oscar winner, or even the observation of the Higgs boson, as proposed by intrade.com, which was liquidated in 2015. As onlinebettingsites.com shows, we could bet on the French elections in 2017, or on the referendum on Brexit (even if for the latter, the predictive markets were not able to reflect the beliefs of the crowds, as an article in The Economist recalls).

Mathématiques du pari-mutuel

The “pari-mutual” theory is not unlike the mutualisation of risks, an important foundation of the insurance mechanism, dear to actuaries. Working in the horse betting markets, Edmund Eisenberg and David Gale obtained, in a short three-page article, Consensus of Subjective Probabilities, relatively general results, as long as the bet is static.

Supposons que I joueurs puissent parier sur J chevaux. Chaque joueur possède une somme totale bi, que l’on normalisera de telle sorte que bi désigne la part de la somme totale misée (et donc b1 +…+ bI =1). Le joueur i peut alors miser la somme bi,j sur le cheval j (avec ici bi,1+…+bi,J = bi). Lorsque les paris sont clôturés, on note pj le montant parié sur le cheval j, autrement dit b1,j+…+bI,j = pj). La contrainte de budget impose que la somme de ces montants soit égale a 1, ce qui donne aux pj une interprétation probabiliste. Nous reviendrons sur l’utilisation de ces « prix » par la suite. On peut aussi noter qj la cote de gain (payoff-odds) définie comme pj-1-1, de telle sorte que pj=(1+ qj) -1. Si on suppose qu’une portion 1-a est gardée par le bookmaker, alors pj= a(1+ qj) -1 et qj =( a -pj)/ pj. Les rendements espérés sur chacun des chevaux doivent être égaux, à l’équilibre, au rendement net attendu, où l’espérance est calculée sous la probabilité p, de manière à refléter les croyances de tous les parieur, soit ici

pjqj+ (1-pj)(-1)= a-1

The key result of the Eisenberg & Gale model is to show that there is a balance in this market. More precisely, the fraction bet on each horse must be equal to the probability of the horse market. To achieve this balance, it is often assumed that the equilibrium ratings are found by an auctioneer (this role will be played by the bookmaker). As Blough (2008) noted, the hypothesis that no wagering is made until the odds are balanced is a hypothesis that is indeed true in horse racing.

If we assume that each bettor is risk neutral (and seeks to maximize his expectation of winning) and that his beliefs are materialized by a probability vectors pi=(pi1,…,piJ) – in the sense that player i thinks that horse j will win with a probability pij – then at equilibrium, if bi,j >0,

pij=pj max{pis/ps}

where argmax{pis/ps}= argmax{pis(qs+1)}

s the horse on which bettor i must bet everything if he bets on a single horse. Blough (2008) elaborates at length on the description of this balance, and extends it to the case where agents potentially have risk aversion (but the same) and potentially different beliefs. This balance is then described as a consensus of belief.

In an article entitled Interpreting the Predictions of Prediction Markets, Charles Manski proposed using this theory to interpret these prices, in conjunction with more traditional approaches in economics, such as Arrow-Debreu prices.

To illustrate this consensus, let us consider a world cup final that should lead either to the victory of A or the victory of B. Let us imagine a contract offering $1 if A wins, and let this contract be offered at price pA. Si on n’autorise pas d’arbitrage, on a une loi du prix unique, et on en déduit que pB = 1-pA. Imaginons un joueur qui pense que la probabilité que A gagne est supérieure à pA, soit, avec les notations précédentes, piA > pA. Dans ce cas, le joueur a intérêt à parier tout son agent sur la victoire de A, c’est-à-dire acheter ce contrat. La demande agrégée pour ce titre sera alors

[b1P[p1A > pA]+…+ bI P[pIA > pA]] / pA

et on aura un équilibre si la demande agrégée pour les deux titres vérifie

[b1P[p1A > pA]+…+ bI P[pIA > pA]] / pA  

= [b1P[p1A < pA]+…+ bI P[pIA < pA]] / pB

 de telle sorte que

pA = b1P[p1A > pA]+…+ biP[piA > pA]] +…+ bIP[pIA > pA]

which allows the prize to be written as an average of the players’ beliefs.

It should be noted here that the balance is static, allowing the bookmaker to just set a rating. Recently, Agrawal et al (2014) proposed an algorithm to balance this market in continuous time. It may also be noted that this notion of equilibrium appears in many algorithms, such as in the so-called Fisher market.

The predictive power of prices

But this idea of seeing in the prices an aggregation of players’ beliefs is not new! In 1655, in Van Rekeningh in Spelen van Gelucken, (published in Latin under the title’De Ratiociniis in Aleæ Ludo’), Christiaan Huyghens proposed to extract information on beliefs from prices. In 1671, Wilhelmina de Witt noted that, as the price of a contract paying an annuity until death could be seen as a weighted average of annuities (with a fixed maturity), by observing the prices of the different insurance contracts, probabilities interpreted as probabilities of survival could be extracted.

These probabilities are “subjective” as Bruno de Finetti or Frank Ramsey called them. The latter did not see probabilities from a frequentist angle, but saw it as a measure of the degree of belief, which could be measured through bets, in Truth and Probability (1926). This is finally what the theory presented by Kenneth Arrow in 1953, and further developed by Gérard Debreu in 1959, introducing the “Arrow-Debreu prices”, says.

Many websites use odds to infer players’ beliefs, which are misrepresented as the probability that a team will win a competition. We can also note the work carried out last summer by doctoral students at the University of Rennes who had compared the odds on online betting sites, and the forecasts obtained by several algorithms (ranging from a naive Bayesian classifier to boosting, SVM or neural networks). A special issue of The Economist, published in 2007, entitled The Future of Futurology, noted that “the most heeded futurists these days are not individuals, but prediction markets, where the informed guesswork of many is consolidated into hard probability”. This idea has now largely returned to the forefront, as predicted in the article by Chen & Pennock (2010) published in AI Magazine.

Agrawal, Shipra, Delage, Erick, Peters, Mark, Wang, Zizhuo & Ye, Yinyu (2014). A Unified Framework for Dynamic Prediction Market Design. Operations Research.

Baron, Ken & Lange, Jeffrey (2006). Parimutuel Applications In Finance: New Markets for New Risks. Springer.

Baumgartner, Frederic (2003) Behind locked doors: a history of papal elections. Palgrave.

Blough, Stephen R. (2008) Differences of opinion at the racetrack. In Efficiency of Racetrack Betting Markets, 323-341, World Scientific.

Chen, Yiling & Pennock, David (2010). Designing Markets for Prediction. AI Magazine.

Decker, Wolfgang & Thuillier, Jean-Paul (2004). Le sport dans l’antiquité. Picard.

Eisenberg, Edmund & Gale, David (1959). Consensus of Subjective Probabilities: The Pari-Mutuel Method. Annals of Mathematical Statistics, 30:1, 165-168.

Griffith, RM (1949) Odds adjustments by American horse-race bettors. The American Journal of Psychology, 62, 290-294.

Manski, Charles (2005) Interpreting the Predictions of Prediction Markets. NBER 10359.

Rhode, Paul, W. & Strumpf, Koleman (2008) Historical Political Futures Markets: An International Perspective. NBER 14377.

[1] Baron & Lange (2006) discusses the comparison between so-called “risk-neutral” valuation in finance (based on the law of single price and arbitrage), and that relating to mutual betting. They thus speak of “self-hedging” because, in a bet, the bettors share the winnings in proportion to their initial bet. This is reminiscent of the way mutual insurance companies operate, where the money used to compensate victims must correspond to the total premiums charged.


A brief history of sports betting

this article was originaly published – in French – in variance.eu

A report by the American Gaming Association (May 2017) estimated that between $100 billion and $400 billion was bet each year on an estimated gross income of between $5 billion and $20 billion, just for sports betting. We will return here to a brief history of sports betting, emphasizing the concept of pari-mutuel betting. We will see, in a second article, the links of this principle with mathematical finance, and insurance.

From games to sports

Sports betting has been around for a long time, even if the origin of the first bet is impossible to date. We can think of the Greeks, inventors of the Olympic Games, where it was not uncommon for spectators to bet among themselves on the winners (Decker & Thuiller, 2004). Closer to home, as Georges Vigarello reminds us, “Under the Ancien Régime, gambling was the subject of a real passion. It takes the form of either betting games or prize games.

The first, bets, are made between people from the same social world, between farmers or between nobles. The second, the prize games, take place during parish celebrations, and show different regional practices, with the struggle in Brittany, or the jump in Provence. We can also think of the confrontations between villages at the soule for example. Among the nobles, prize games are organized for special occasions (birth or wedding). These games were recreational and festive moments.

It was not until the end of the 19th century that gambling became a sport, in line with the hygienist theories of the time. We can think of Baron Pierre de Coubertin, who wanted to “use all the means appropriate to develop our physical qualities to make them serve the collective good” through “sport”. We will find the Baron again in 1887 with the creation of the Union of French Societies of Athletic Sports, the official appearance of the notion of “sport”, replacing that of “game”, as Dietschy & Clastres (2006) points out, noting in passing that this Union is based on amateurism, in reaction against the companies of cycling (from 1860) and walking (around 1870) which resumed the traditions of price and betting games. Around 1890, this union, dedicated to athletics, opened up to other sports (rugby, field hockey, fencing, swimming) which were represented by specialized commissions.

The first bookmakers and gambling

A little earlier, during the Industrial Revolution, horse betting organised by bookmakers developed. These bets were popular in the United Kingdom in the 16th and 17th centuries, but remained reserved for the aristocracy and the landed gentry. And in reality, only horse owners were allowed to bet on the results of these private races, known as “matches”. One of his races, launched by the twelfth Earl of Derby (Edward Smith-Stanley) around 1870, also left its mark on sporting vocabulary. If these races were originally private, Charles II’s passion for these races made them more ambitious, attracting huge crowds, betting more and more important sums. Innkeepers and pub owners were then major promoters of these races, which encouraged owners to organize the races near their establishments. They then naturally became the first bookmakers, organizing the first steeple-chases, a form of race (first created in Ireland) where riders ran from one church tower to another by jumping everything in their path! In 1826, at the stables in Saint Alban, north of London, the idea of horses starting and finishing in the same place was launched, giving rise to modern racecourses.

Betting was not yet regulated and betting on races was based on a credit system. And since gambling near a place where alcohol was available in large quantities can have dramatic consequences, the British government banned gambling in pubs, which led to the opening of betting shops, run by bookmakers, with the adoption of the Gambling Act in 1845. The bookmakers not only played the role of scribes, keeping track of transactions in registers, they also served as arbitrators in betting. The bookmaker has become the intermediary with whom to bet, he receives the bets, but does not bet against the player. The arbitrator does not only act at the end, in the event of a dispute, but above all to make the bet official. Indeed, cash bets were rare, and bookmakers decided whether the items bet had the same value and, if not, what the difference was. One of the players then adds money to a cap. Players put their hands in the hat and remove them, either to agree with the assessment or to indicate their disagreement. This is referred to as “hand in cap”, which refers to the amount of money needed to ensure a fair bet. The word “handicap” was then commonly used in horse betting (to designate disadvantaged participants at the start of a race) and then to have a medical connotation from 1950 onwards.

Thereafter, bookmakers will not lack imagination, introducing cash bets, then offering fixed odds against each horse in a race. Parliament then went backwards with the Suppression of Betting Houses Act in 1853. Credit institutions and games of chance on racetracks were allowed. At the same time, in France, Léon Sari invented the “pari mutuel” in 1857 with Charles de Morny, owner of the Maisons-Laffitte racetracks (which became a building with stands in June 1878). Joseph Oller, who co-founded the Moulin-Rouge, is the concessionaire. As the Senate report on gambling in France reminds us, the law of June 2, 1891 legalizes betting on horse races and establishes the principle of mutualization. As we will see later, this principle means that bettors play against each other and share the winnings (once the legal levies provided for by law have been made for the benefit of the State and the institution of racing). In mathematical finance, we speak of “self-hedging strategy”. In March 1931, the PMU (“pari mutuel urbain”) was born, and it was not until 1985 that the “sports lotto” arrived.

From horses to other sports

The “pool” has long referred in England to a game of cards played for collective stakes, drawing its etymology from the French “hen”, or rather from the old French “hen”, referring to a young poultry (we will find the Latin word pulla, de pullus, the “young animal”), but also “booty” or “looting”. Here we find the idea of playing for money. This use can be traced back to 1870 (in the sense of “collective betting”) before becoming a pool during the First World War, and then to designate a group of people sharing skills. As early as 1920, the term “football pool” was coined, as recalled by Forrest (1999).

In Liverpool, England, John Moores founded Littlewoods in 1923, a retail company, before launching mail order sales, while offering football bets. The most famous game was the “Treble Chance”, where players could choose to bet on 10, 11 or 12 football matches for the coming weekend. Anecdotally, as noted by Forrest & Pérez (20013), when a match could not take place (for example because of rain), a panel of experts appointed by Littlewoods had to model the match, and provide a forecast. After the Second World War, in Europe, we will see the arrival of so-called 1X2 formulas where the player must predict whether, in a set of 12 to 15 games, the home team will win (1), lose (2) or draw (X). It can be noted that these “football pools” could refer to any form of pari-mutuel betting, very strongly resembling a lotto. The main difference is that in the lottery, the draw is supposed to be a pure random process, unlike football matches. And for the players, the difference is significant! In the 1980s, Liverpool was one of the largest private companies in Europe. Before decreasing with the birth of online betting sites….

Internet and online betting

Now, in addition to the betting companies that still exist in the United Kingdom, the strong point of bookmakers is their online presence. The first sites were created around 1995, for example Intertops, which was based on a law passed by the island nation of Antigua and Barbuda (an officially independent, Commonwealth member country) in 1994, granting licences to companies wishing to provide gambling services over the Internet (subsequently, they obtained licences from the Mohawk territory of Kahnawake in Quebec, or Malta). Betting on sports events has quickly become very popular.

In 2000, Betfair was launched, and revolutionized the industry: Betfair itself did not take customer bets, but rather offered customers to place bets between them. These peer-to-peer betting was quickly very popular. In 2002, the first live betting was launched, offering bettors the opportunity to bet on a sporting event while it was taking place. Today, on lƒes larger sites, all kinds of sports are available, whether collective (football, basketball) or individual (tennis, boxing), with possibly a competition involving more than two players or teams (athletics, cycling). The player can choose an objective, which can be a final score (1X2 in football), a number of goals scored, etc., then he concludes the bet by choosing the amount he is willing to bet (the bet). On all sites, no less than 20,000 bets are possible every day.

Decker, Wolfgang & Thuillier, Jean-Paul (2004). Le sport dans l’antiquité. Picard.

Dietschy, Paul & Clastres, Patrick (2006). Sport, société et culture en France du XIXe siècle à nos jours. Hachette, Carré Histoire.

Forrest, David (1999). The Past and Future of the British Football Pools. Journal of Gambling Studies, 15:2, 161-176.

Forrest, David & Pérez, Levi (2013) The Football Pools in The Oxford Handbook of the Economics of Gambling, 147-162

Vigarello, Georges (2004) Le sport est-il encore un jeu ? Sciences Humaines, no 152.

To be continued…. with a post on how bets, predictions and players’ beliefs are linked.

Tout le monde peut sortir vainqueur d’un pari ?

Petit paradoxe du jour (emprunté à http://www.futilitycloset.com/). Avec un collègue, suite à un pari stupide, nous nous sommes engagé à mettre une cravate, et celui qui a payé sa cravate le plus cher devra la donner à l’autre (ce que nous ne savions pas avant de choisir ladite cravate). A priori, nous avons autant de chance l’un que l’autre de gagner. Si je perds, je perds le prix de ma cravate, disons http://freakonometrics.blog.free.fr/public/perso5/cravate01.gif. Alors que si je gagne, je gagne une cravate qui vaut plus cher que la mienne, disons http://freakonometrics.blog.free.fr/public/perso5/corrrec-01.gif, avec http://freakonometrics.blog.free.fr/public/perso5/corrre-2.gif. Aussi, mon espérance de gain est


aussi, en moyenne, je vais sortir gagnant de ce pari. Sauf que mon collègue peut faire le même raisonnement… Donc en moyenne, on y gagne tous… Étonnant, non ?

Foot, probabilité, et assurance, partie 2

Continuons le billet commencé ici, où j’essayais noté que les enseignes qui se sont engagé à faire des cadeaux en cas de victoire de l’équipe de France s’étaient couvertes via des contrats d’assurance. En fait, elles peuvent également se couvrir sur les marchés (en l’occurrence les sites de paris en ligne). Il existe plusieurs façons de faire des paris. La première est de faire (ou d’acheter) un pari sur un événement (la France ira en finale de la coupe du monde, par exemple). Ce pari est assorti d’une cote que défini le bookmaker. A une cote de 20 contre 1, si je paris 1 euro, je gagne 20 euros en cas de qualification pour la finale.
Dans ce cas particulier, le bookmaker est un agent important dans le marché, car il fixe le prix. Le site oddschecker.com (ici) permet de voir les cotes offertes par différents bookmakers.

Mais il existe aussi des marchés de paris, sans bookmakers. Il s’agit réellement d’un marché, où les parieurs prennent des positions les uns contre les autres, le bookmaker ne servant que d’intermédiaire et d’animateur sur ce marché. C’est le cas sur betfair.com () qui est un site d’échange de paris.

Pour commencer simplement, imaginons que nous sommes quelques jours avant une finale opposant la Corée du Sud aux Etats-Unis. Comme il ne peut y avoir qu’un vainqueur il existe des contratssymétrique, où l’on peut gagner 20 euros si la Corée gagner (et rien si elle perd) et un autre où l’on peut gagner 20 euros si la Corée perd. Bref, soit je suis acheter sur la Corée, soit je suis vendeur. Le prix de ces contrats indiquent la probabilité qu’a la Corée de gagner. On parle aussi de back or lay, deux évènements étant possibles.

  • Lecture d’une cote

Regardons maintenant le match d’ouverture, opposant l’Afrique du Sud au Mexique. Les cotes sont les suivantes, “Mexique 6/4, Afrique du Sud 15/8 et Draw (match nul) 11/5” sur un des sites. Le premier nombre indique ce que l’on gagne (net) en misant le second, si mon pari est gagnant. Autrement dit, si je paris 5 euros sur un match nul, et que match nul il y a, alors mon bénéfice sera de 11 (que je touche en plus de ma mise). La cote est alors de 1+11/5, soit 16/5 (j’avais fait ici un billet sur le lien entre cote et probabilité). Elle indique non plus le bénéfice, mais le gain: en misant 5 euros, j’empoche 16 euros. On parlera de paris 1X2.

Pour aller plus loin, on peut aussi parier sur le nombre de buts marqués. On parlera de paris over/under.

Chez le même bookmaker, si je paris 1 euro que la France gagne la coupe du monde, je fais un bénéfice de 18 euros.

  • Cote ou probabilité ?

Lorsque j’avais essayé d’expliquer l’origine de la loi logistique (ici), j’avais expliqué qu’il existait un lien simple entre probabilité et cote, la cote étant la probabilité que l’évènement se réalise, divisé par la probabilité qu’il ne se réaliste pas. Malheureusement, les choses sont un peu plus complexes dans les paris…
Zoomons un peu sur le match d’ouverture,

La cote est la valeur offerte la par le bookmarker, à laquelle on ajoute 1. La “probabilité” est alors l’inverse de cette cote.

résultat Bookie Offers Cote “probabilités”
Mexico 11/8 2.375 42.10%
South Africa 13/8 2.623 38.09%
Draw 11/5 3.200 31.25%
total 111.45%

A-t-on raté quelquechose ? car visiblement la “probabilité” n’est pas une vraie probabilité… En fait, supposer que l’on puisse construire une mesure de probabilité à partir des cotes doit pouvoir être relié au “théorème fondamental d’asset pricing” (et de probabilités risques neutres mentionnés tout à l’heure)

  • Relecture des cotes à l’aide du théorème fondamental

Autrement dit, on suppose qu’il existe une mesure http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-01.png telle que
http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-02.pnghttp://freakonometrics.hypotheses.org/files/2017/03/foot-pari-03.png est l’ensemble des gains possibles, et http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-04.png un équivalent certain.
On suppose qu’il existe un ensemble d’évènements http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-05.png (mutuellement exclusif comme on dit, formant une partition de http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-06.png), un seul pouvant survenir (en l’occurrence l’équipe qui reçoit gagne, ou l’équipe qui reçoit perd, ou il y a match nul). On considère un bookmaker qui accepte de payer http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-08.png si l’évènement i survient à un joueur qui aura parié 1 euro. On pourra légitimement pense que http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-09.png. Notre souhait est de relier ces http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-08.png et http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-10.png, si un lien quelconque pouvait exister. Ce problème avait été soulevé par Ramsey ou de Finetti.

En notant http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-11.png le montant total placé sur l’évènement http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-12.png. On pourra dire que l’on est à l’équilibre si http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-13.png est constant. Autrement dit, le ratio des cotes doit être l’inverse des ratios de montants placés,

http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-15.pngOn peut ainsi s’intéresser à


qui est la fraction de l’argent qui sera effectivement rétribuée aux parieurs, ou http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-17.png sera la part gardée par le bookmaker. On peut penser qu’un bookmaker fixe http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-18.png, et alors


En fait, k peut être vu comme un taux d’actualisation (l’équivalent d’un http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-22.pngen mathématique financière). On peut d’ailleurs noter que si on note http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-25.png le produit


alors la stratégie consistant à parier


sur l’évènement http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-12.png, pour tout les évènements, aura un coût unitaire (j’ai ici constitué mon portefeuille si on revient à la théorie de l’arbitrage). Aussi


En l’occurrence, sur le match d’ouverture de la coupe du monde, on obtient http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-18.png de l’ordre de 90% (l’inverse de la somme de la dernière colonne). Autrement dit, le bookmaker prend de l’ordre de 10%.
A partir de là, on peut en déduire de ces prix d’Arrow Debreu une “probabilité implicite” qui est la probabilité risque neutre, ou la croyance des investisseurs dans la survenance des évènements. Pour cela, on pourrait poser


sauf que dans ce cas, compte tenu de la dernière relation, http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-34.png. On définie alors la probabilité implicite des parieurs sous la forme


On a réussit à construire une mesure de probabilité http://freakonometrics.hypotheses.org/files/2017/03/foot-pari-01.png, qui correspond à l’intuition des parieurs… mais qui n’a rien à voir avec la probabilité réelle. Si l’on compare les bookmakers, on note que les probabilités induites sont très proches,

et si on compare la part prise par le bookmaker sur ce match, on est là aussi sur des ordres de grandeurs comparables,

Justin Wolfers et Eric Zitzewitz, dans Prediction Markets paru dans le Journal of Economic Perspectives de 2004 (ici) expliquent d’ailleurs tout cela très clairement. Je peux aussi renvoyer à un document de travail datant de novembre dernier de Egon Franck, Erwin Verbeek, et Stephan Nüesch (en ligne ici).
J’ai parlé tout à l’heure de probabilité risque neutre, mais cela sous-entend que l’on puisse supposer les marchés efficients, et qu’il n’y ait absence d’opportunité d’arbitrage. C’est précisément ce qu’avait regardé Steven Levitt et Richard Gillici, dans un vieux papier, évoqué également . Pour résumer les conclusions, il n’y a pas d’opportunités d’arbitrage sur ces marchés, et il existe des “market makers”, c’est à dire des parieurs misant énormément, ce qui accroît considérablement la liquidité de ces marchés. Mais cette conclusion sur l’absence d’opportunité d’arbitrage a été largement remise en cause en pratique. Pour aller plus loin, on peut aussi regarder un article de Moris Eaton et David Freedman, Dutch Book against some ‘Objective’ Prior, paru dansBernoulli en 2004, en ligne . C’est marrant car cette relecture Bayésienne des jeux de pari peut être étudier en pratique, pour la coupe du monde de foot surworldcup.bayesialab.com (, merci Avner pour le lien). Je vais continuer de suivre ça lorsque les matchs commençerons, afin de voir les mises à jour des paris au fur et à mesure de l’évolution des matchs… à suivre donc.