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.

 

La valeur de la vie

Un court article, écrit conjointement avec Béatrice Cherrier… Tous les commentaires sont les bienvenus !

En 1928, revenant sur la révolution chinoise de 1925, André Malraux publie son roman Les Conquérants, et glisse « j’ai appris qu’une vie ne vaut rien, mais que rien ne vaut une vie ». Si la formule peut plaire, on imagine qu’elle n’aidera pas trop un décideur public. En 2013, le Commissariat Général à la stratégie et à la prospective, en France, évaluait la valeur d’une vie à 3 millions d’euros. Mais d’où sort ce chiffre ? Et que signifie-t-il vraiment ?

Sauver une vie vaut-il le coup?

Chiffrer une vie est en effet un problème auxquels les assureurs, mais aussi les décideurs publics sont confrontés bien plus souvent qu’il n’y paraît. Après l’effondrement du World Trade Center en 2011, le Congrès américain adopta la Loi sur la sécurité du transport aérien et la stabilisation des systèmes. Cette nouvelle loi prévoyait la création d’un fonds spécial pour indemniser les victimes des attentats du 11 septembre 2001. Le montant de l’indemnité, et les personnes qui y auraient droit, seraient décidées par un fonctionnaire tout-puissant. Ce « special master,» Kenneth Feinberg, explique dans un ouvrage qui revient sur son expérience (Feinberg 2006) que le gouvernement souhaitait éviter une avalanche de poursuites pour préjudices corporels. Cela aurait pu plonger l’industrie du transport aérien dans la tourmente.

Un cadre très strict fut donc instauré : seules les victimes « ayant recu un traitement à l’hopital dans les 72 heures qui suivirent les attaques », blessées aux abords du World Trade Center et du Pengatone, ainsi que leurs conjoints et enfants – mais pas leurs parents – furent déclarés admissibles à une indemnisation. Le fond accorda plus de 7 milliards de dollars à 5 560 victimes et membres de leur famille. Feinberg se devait, légalement, d’étalonner les dommages et intérêts en fonction de la « valeur financière » de la victime décédée. Il dû ainsi expliquer à la femme d’un pompier, par exemple, que son mari valait moins qu’un courtier d’assurance.

En France, le récent passage aux 80km/h sur les routes à deux voies fut aussi partiellement justifié par les vies sauvées. Alors que le premier ministre se réjouissait, en janvier, d’un bilan de 116 vies épargnées, la journaliste Alba Ventura (2019) s’interroge sur RTL : « s’il s’agit de ne sauver qu’une vie, est-ce que ça ne vaut pas le coup ? » Le support radiophonique ajoute à l’ambigüité. Demande-t-elle en fait si « cela ne vaut pas le coût? »  Car le problème est bien celui des méthodes employées par la puissance publique pour chiffrer le prix d’une vie, sauvée ou perdue.

La valeur d’une vie humaine comme taux marginal de substitution

A la fin des années 1940, l’US Air Force cherchait à maximiser les dommages infligés par ses raids aériens contre l’Union Soviétique. Quand un groupe de chercheurs de la RAND Corporation proposa de faire voler un grand nombre d’avions peu coûteux pour leurrer les défenses aériennes soviétiques, les généraux de l’Air Force refusèrent l’idée, arguant que le coût de la vie des pilotes sacrifiés ne figurait pas dans les calculs. Comme le rappelle Spencer Banzhaf (2014),  l’économiste de la défense Jack Hirshleifer choisit alors de calculer valeur de la vie d’un pilote en intégrant le coût de sa formation et de son remplacement. Cette réponse avait l’avantage d’utiliser des grandeurs directement monétaires, et facilement quantifiables. Dans les années 1960, sur l’influence des réflexions autour du capital humain, il fut suggéré d’utiliser une estimation des salaires nets actualisés perçus au cours d’une vie de pilote, supposés refléter l’utilité matérielle du métier. Ces méthodes restaient dans la lignée de celles définies par Louis Dublin et Alfred Lotka pour les compagnies d’assurance dans l’entre-deux-guerres (Cavalin 2016).

S’il reprend comme titre le slogan publicitaire d’une compagnie d’assurance, popularisé par des organismes de sécurité routière ( « The Life You Save May Be Your Own »),Thomas Schelling, prix Nobel d’économie en 2005, publie en 1968 un article qui rompt largement avec cette tradition.  Il utilise en fait le travail de l’un de ses étudiants (et ancien pilote militaire) Jack Carlson, qui cherchait à évaluer si lcertains investissements en matière de sécurité (pour les pilotes) « valaient le coût ». Le coût d’un système d’éjection des avions B-58 était par exemple de l’ordre de 80 000 dollars, pour un gain substantiel sur la probabilité de survie. C’est cette idée de lier la valeur de la vie avec la notion de risque qui permis à Schelling de développer le concept de « valeur statistique » de la vie.

L’innovation majeur de Schelling consistait à impliquer les citoyen.ne.s dans l’évaluation de la valeur de leur propre vie. Puisqu’il était stérile de leur demander de chiffrer leur propre vie de but en blanc, on pouvait en révanche adapter la méthode de Carlson en leur  demandant, par exemple, combien ils et elles seraient prêtes à dépenser pour un airbag, ou un traitement médical, qui diminuerait leur taux de mortalité de 1%. Ainsi, en se plaçant dans un diagramme représentant en abscisse la probabilité de survie où l’espérance de vie résiduelle et en ordonnée la richesse, comme sur la Figure 1, pouvait-on construire des courbes d’indifférence liant richesse et survie : quelle somme accepte-t-on (marginalement) de dépenser pour gagner statistiquement un peu de vie, soit en diminuant sa probabilité de décès, soit en allongeant son espérance de vie ? Dans l’exemple ci-dessous, la valeur de la vie est simplement la dérivée de la courbe d’indifférence.

Figure 1 : Arbitrage entre espérance de vie et richesse.

La valeur de la vie n’est alors pas une grandeur constante, mais dépend de la situation dans laquelle on se trouve. Aussi,SVL=\frac{d\omega}{d\text{E}}=\frac{d\omega}{d\text{p}}suivant qu’on la calcule par rapport à une variation de l’espérance de vie, ou de la probabilité de décès. Si on a une espérance de vie résiduelle plus ou moins grande (à gauche ou à droite), ou si on est plus ou moins riche (en haut ou en bas), la pente ne sera pas la même. Un exemple classique est celui d’une roulette russe, avec un pistolet a douze chambres. Supposons qu’il y a 3 balles, quel serait le montant que l’on serait prêt à payer pour enlever une balle ? Que deviendrait ce montant s’il y avait 9 balles et qu’on souhaite en enlever plusieurs ? Supposons que la valeur statistique de la vie soit de 3 millions d’euros. Dans le premier cas, la probabilité passe de 3/12 à 2/12, soit dp_1=1/12 (soit une baisse de 1/3). Dans le second cas, pour avoir aussi une baisse de 1/3, il faudrait passer de 9/12 a 6/12, soit dp_2=3/12. Si on suppose que la valeur statistique de la vie est constante, alors d\omega_2/d\omega_1=d\text{p}_2/d\text{p}_1=3, et on devrait être prêt à dépenser 3 fois plus pour une même baisse relative de probabilité de décès.

D’un point de vue heuristique, dans le second cas, on est dans une situation un peu désespérée (on a 3 chances sur 4 de mourir) et donc toute solution est bonne à prendre, quelle que soit son prix ! C’est ce que l’on retrouve au travers de la convexité de la courbe de droite sur la Figure 1 : si ma probabilité de décès est élevée (à droite sur l’axe des abscisses), je suis prêt à dépenser beaucoup, pour un faible gain. Cette manière d’évaluer sa propre vie, proposée par Schelling, est souvent, appelée « gunpoint value.»

Sauver ma vie, ou celle d’autrui ?

Mais cette approche répond-elle vraiment à la question de départ ? La vie sauvée par une mesure de sécurité contraignante et couteuse est rarement celle de la personne qui prend une décision. Cette tension est particulièrement visible au sein des débats français sur la mesure de la valeur d’une vie, puisqu’à la différence des Etats-Unis celle-ci est largement le fait d’ingénieurs-économistes recrutés par l’Etat afin de mettre en place des politiques publiques visant a augmenter le bien-être des populations. La question de la sécurité routière est à l’origine d’un article fondateur sur le sujet, présenté par deux ingénieurs des ponts et chaussées, Claude Abraham et Jacques Thedie, au colloque annuel de recherche opérationnelle d’Aix-en-Provence en 1960.

Répondant à la question « combien une collectivité doit dépenser pour sauver une vie humaine », ils pointent deux types d’éléments à quantifier. Les éléments « objectifs » de nature « économique, » quantifiable en actualisant les pertes de salaires directes et les pertes de production et de consommation, grâce un raisonnement pragmatique qui mélange capital humain et analyse macroéonomique. Par exemple, un homme de 41-45 ans a une valeur de production deux fois supérieure à un homme de 56-60 ans, et sa valeur de consommation est 50% plus élevée. Mais la perte d’un homme de plus de 65 ans est en réalité un gain, ce qui montre l’importance d’intégrer les éléments « affectifs.» Puisque leur évaluation est autrement plus difficile, ceux-ci s’en remettent à l’estimation faite par les tribunaux en matière d’indemnisation des dommages personnels, en particulier l’attribution d’un praetium doloris.

Comme le relate Daniel Benamouzig (2005), les aspects théoriques, techniques, éthiques et métaphysiques du principe et de la méthode de la quantification d’une valeur de vie présentés par Abraham et Thédié, et en particulier de l’application de telles méthodes dans le champ de la santé, firent l’objet de débats houleux. Ceux-ci n’ont d’ailleurs toujours pas fait l’objet de résolution consensuelle. Françoise Favre (1970) note par exemple que l’utilisation du calcul économique de la valeur d’une année-vie pour décider si un dépistage systématique du cancer du col de l’utérus doit être mis en place peut conduire par construction une réponse négative. En effet, la valeur marchande du travail féminin qui sert de base au calcul est largement inférieur à celle du travail masculin, ce qui crée des inégalités de traitement hommes-femmes.

Adoptant un cadre éthique et théorique empruntant au choix social, Jacques Drèze (1962) propose une méthode de calcul alternative plus proche de celle développée par Schelling. Une décision publique doivent se fonder sur les préférences de la collectivité issues de l’agregation des utilités individuelles pour la décision en question. Une solution aux problèmes de mesure et d’incommensurabilités soulevé par Abraham et Thédié consiste a poser la question aux citoyens « combien la collectivité doit-elle dépenser, selon vous, pour sauver une vie? » L’utilité de la vie peut-être calculée en identifiant la disposition individuelle subjective à payer pour prolonger sa vie en écartant un risque déterminé, ajoute Drèze. Celui-ci conclue que sa méthode conduit à une estimation de la valeur de vie nettement supérieure à celle à laquelle aboutissent ces collègues. La sensibilité des évaluations aux méthodes de calcul, est, aujourd’hui encore, un problème majeur.

Plusieurs méthodes, plusieurs valeurs ?

Biausque (2011) reprend plusieurs études, afin d’estimer la valeur (statistique) de la vie, face à des risques environnementaux, de sante ou routier que l’on peut résumer dans le Tableau 1.

Environnement Santé Trafic routier
Nb d’études 51 250 65
Moyenne (€) 2 455 982 2 574 149 4 884 853
Minimum (€) 24 427 4 450 267 615
Maximum (€) 7 641 706 22 100 000 17 500 000

Tableau 1 : source Biausque (2011).

On voit que ces calculs sont complexes, et donnent lieu à des ordres de grandeurs très différents les uns des autres. La variabilité entre individus était évoquée dans Feinberg (2006) qui expliquait qu’il pouvait être économiquement juste à dire que la vie d’un trader de 25 ans “valait plus” qu’un pompier de 45 ans. Mais c’est surtout la variabilité entre les méthodes, que l’on retrouve également dans Hugonnier et al. (2018) qui surprend, et dérange, avec un facteur allant de 1 à 20 suivant la méthode utilisée.

état de santé Quintile

0%-20%

Quintile

40%-60%

Quintile

80%-100%

Statistique ‘fair’ 4 380 000 4 400 000 7 890 000
‘very good’ 8 800 000 8 830 000 12 135 000
Gunpoint ‘fair’ 235 000 235 000 422 000
‘very good’ 590 000 590 000 650 000
Capital humain 250 000 390 000 525 000

Tableau 2 : source Hugonnier et al. (2018)

Le Tableau 2 reprend la valeur statistique (inspirée de Drèze 1962), celle basée sur des calculs de capital humain, ainsi qu’une « gunpoint value », en fonction du niveau de richesse de la personne qui décède (niveaux de quantiles) et de l’état de santé de la personne (avant son décès).

Ces tableaux montrent à quel point il est difficile d’évaluer la vie de personnes impliquées dans un accident mortel. On essaye d’imaginer la valeur de la vie d’un « individu représentatif » (peut être en fonction de son état de santé, de son âge, de son revenu). Mais comment faire pour attribuer une valeur à une vie qui n’existe pas encore ? Car nombre de décisions prises aujourd’hui impactant aussi les « générations futures », c’est-à-dire des personnes qui aujourd’hui n’existent pas… Est-il possible de donner une valeur à la vie de ces personnes ? Car c’est normalement ce qu’il convient de faire si on veut mettre en place une politique visant à limiter le réchauffement climatique.

References

Abraham, С. & Thedié, J. 1960 Le prix d’une vie humaine dans les décisions économiques. Revue française de Recherche opérationnelle. 16 : 157-168

Banzhaf, Spenser H. 2014. Retrospectives: The Cold-War Origins of the Value of Statistical Life. Journal of Economic Perspective, 28 :4, 213-226.

Benamouzig, Daniel. 2005. La Santé au miroir de l’Economie. Paris : PUF

Biausque V. 2011, Valeur statistique de la vie humaine : une méta-analyse. OCDE

Cavalin, C. 2016. « La valeur d’une vie statistique : histoire américaine, histoire de la pensée économique. » Incidence 12.

Commissariat général à la stratégie et à la prospective 2013. Éléments pour une révision de la valeur de la vie humaine. http://www.strategie.gouv.fr/

Costa, Dora L. & Kahn Matthew E. 2004. Changes in the value of life, 1940-1980. Journal of Risk and Uncertainty, 29 :2, 159-180

Drèze, Jacques 1962. L’utilité sociale d’une vie humaine. Revue française de recherche opérationnelle 23 : 3 -28

Fabre, Françoise. 1970. « Une étude économique de la prévention et du dépistage précoce du cancer du col de l’utérus » Cahiers du Séminaire d’Econometrie 12, 121-143

Feinberg, Kenneth R. 2006. What Is Life Worth?: The Inside Story of the 9/11 Fund and Its Effort to Compensate the Victims of September 11th. Public Affairs.

Johansson, Per-Olov, 2000. Is there a meaningful definition of the value of statistical life? Journal of Health Economics, 20, 131-139

Hugonnier, J., Pelgrin, F. & St-Amour, P. 2018. Valuing Life as an Asset, as a Statistic and at Gunpoint. Swiss Finance Institute Research Paper 18-27

Lery, Simon 2004. Arbitrages : le prix de la vie. Alternatives Economiques, 223.

Mrozek, Janusz R. & Taylor, Laura O. 2002. What determines the value of life : a meta analysis. Journal of Policy Analysis and Management, 21 :2, 253–270

Schelling, T.C. 1968. ‘The life you save may be your own.’ In Problems in PublicExpenditure Analysis ed. Samuel B. Chase (Washington DC: Brookings Institution) 127–162

Ventura, Alba. 2019. 80km/h : « S’il s’agit de ne sauver qu’une vie, est-ce que ça ne vaut pas le coup ? », RTL, 29 janvier 2019.

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.

What it the interpretation of the diagonal for a ROC curve

Last Friday, we discussed the use of ROC curves to describe the goodness of a classifier. I did say that I will post a brief paragraph on the interpretation of the diagonal. If you look around some say that it describes the “strategy of randomly guessing a class“, that it is obtained with “a diagnostic test that is no better than chance level“, even obtained by “making a prediction by tossing of an unbiased coin“.

Let us get back to ROC curves to illustrate those points. Consider a very simple dataset with 10 observations (that is not linearly separable)

x1 = c(.4,.55,.65,.9,.1,.35,.5,.15,.2,.85)
x2 = c(.85,.95,.8,.87,.5,.55,.5,.2,.1,.3)
y = c(1,1,1,1,1,0,0,1,0,0)
df = data.frame(x1=x1,x2=x2,y=as.factor(y))

here we can check that, indeed, it is not separable

plot(x1,x2,col=c("red","blue")[1+y],pch=19)

Consider a logistic regression (the course is on linear models)

reg = glm(y~x1+x2,data=df,family=binomial(link = "logit"))

but any model here can be used… We can use our own function

Y=df$y
S=predict(reg)
roc.curve=function(s,print=FALSE){
  Ps=(S&gt;=s)*1
   
  FP=sum((Ps==1)*(Y==0))/sum(Y==0)
     
  TP=sum((Ps==1)*(Y==1))/sum(Y==1)if(print==TRUE){print(table(Observed=Y,Predicted=Ps))}
   
vect=c(FP,TP)names(vect)=c("FPR","TPR")return(vect)}

or any R package actually

library(ROCR)

perf=performance(prediction(S,Y),"tpr","fpr")

We can plot the two simultaneously here

plot(performance(prediction(S,Y),"tpr","fpr"))
V=Vectorize(roc.curve)(seq(-5,5,length=251))points(V[1,],V[2,])segments(0,0,1,1,col="light blue")

So our code works just fine, here. Let us consider various strategies that should lead us to the diagonal.

The first one is : everyone has the same probability (say 50%)

S=rep(.5,10)plot(performance(prediction(S,Y),"tpr","fpr"))

V=Vectorize(roc.curve)(seq(0,1,length=251))points(V[1,],V[2,])

Indeed, we have the diagonal. But to be honest, we have only two points here : (0,0) and (1,1). Claiming that we have a straight line is not very satisfying… Actually, note that we have this situation whatever the probability we choose

S=rep(.2,10)plot(performance(prediction(S,Y),"tpr","fpr"))

V=Vectorize(roc.curve)(seq(0,1,length=251))points(V[1,],V[2,])

We can try another strategy, like “making a prediction by tossing of an unbiased coin“. This is what we obtain

set.seed(1)

S=sample(0:1,size=10,replace=TRUE)plot(performance(prediction(S,Y),"tpr","fpr"))

V=Vectorize(roc.curve)(seq(0,1,length=251))points(V[1,],V[2,])segments(0,0,1,1,col="light blue")

We can also try some sort of “random classifier”, where we choose the score randomly, say uniform on the unit interval

set.seed(1)

S=runif(10)plot(performance(prediction(S,Y),"tpr","fpr"))

V=Vectorize(roc.curve)(seq(0,1,length=251))points(V[1,],V[2,])segments(0,0,1,1,col="light blue")

Let us try to go further on that one. For convenience, let us consider another function to plot the ROC curve

V=Vectorize(roc.curve)(seq(0,1,length=251))

roc_curve=Vectorize(function(x) max(V[2,which(V[1,]&lt;=x)]))

We have the same line as previously

x=seq(0,1,by=.025)

y=roc_curve(x)lines(x,y,type="s",col="red")

But now, consider many scoring strategies, all randomly chosen

MY=matrix(NA,500,length(y))for(i in 1:500){
  
S=runif(10)
  
V=Vectorize(roc.curve)(seq(0,1,length=251))
  
MY[i,]=roc_curve(x)}plot(performance(prediction(S,df$y),"tpr","fpr"),col="white")for(i in 1:500){lines(x,MY[i,],col=rgb(0,0,1,.3),type="s")}lines(c(0,x),c(0,apply(MY,2,mean)),col="red",type="s",lwd=3)segments(0,0,1,1,col="light blue")

The red line is the average of all random classifiers. It is not a straight line, be we observe oscillations around the diagonal.

Consider a dataset with more observations


myocarde = read.table("http://freakonometrics.free.fr/myocarde.csv",head=TRUE, sep=";")

myocarde$PRONO = (myocarde$PRONO=="SURVIE")*1

reg = glm(PRONO~.,data=myocarde,family=binomial(link = "logit"))

Y=myocarde$PRONO

S=predict(reg)plot(performance(prediction(S,Y),"tpr","fpr"))

V=Vectorize(roc.curve)(seq(-5,5,length=251))points(V[1,],V[2,])segments(0,0,1,1,col="light blue")

Here is a “random classifier” where we draw scores randomly on the unit interval

S=runif(nrow(myocarde)plot(performance(prediction(S,Y),"tpr","fpr"))

V=Vectorize(roc.curve)(seq(-5,5,length=251))points(V[1,],V[2,])segments(0,0,1,1,col="light blue")

And if we do that 500 times, we obtain, on average

MY=matrix(NA,500,length(y))for(i in 1:500){
  
S=runif(length(Y))
  
V=Vectorize(roc.curve)(seq(0,1,length=251))
  
MY[i,]=roc_curve(x)}plot(performance(prediction(S,Y),"tpr","fpr"),col="white")for(i in 1:500){lines(x,MY[i,],col=rgb(0,0,1,.3),type="s")}lines(c(0,x),c(0,apply(MY,2,mean)),col="red",type="s",lwd=3)segments(0,0,1,1,col="light blue")

So, it looks like me might say that the diagonal is what we have, on average, when drawing randomly scores on the unit interval…

I did mention that an interesting visual tool could be related to the use of the Kolmogorov Smirnov statistic on classifiers. We can plot the two empirical cumulative distribution functions of the scores, given the response Y

score=data.frame(yobs=Y,
                 ypred=predict(reg,type="response"))

f0=c(0,sort(score$ypred[score$yobs==0]),1)

f1=c(0,sort(score$ypred[score$yobs==1]),1)plot(f0,(0:(length(f0)-1))/(length(f0)-1),col="red",type="s",lwd=2,xlim=0:1)lines(f1,(0:(length(f1)-1))/(length(f1)-1),col="blue",type="s",lwd=2)

we can also look at the distribution of the score, with the histogram (or density estimates)

S=score$ypred

hist(S[Y==0],col=rgb(1,0,0,.2),
     probability=TRUE,breaks=(0:10)/10,border="white")hist(S[Y==1],col=rgb(0,0,1,.2),
     probability=TRUE,breaks=(0:10)/10,border="white",add=TRUE)lines(density(S[Y==0]),col="red",lwd=2,xlim=c(0,1))lines(density(S[Y==1]),col="blue",lwd=2)

The underlying idea is the following : we do have a “perfect classifier” (top left corner)

is the supports of the scores do not overlap

otherwise, we should have errors. That the case below

we in 10% of the cases, we might have misclassification

or even more missclassification, with overlapping supports

Now, we have the diagonal

when the two conditional distributions of the scores are identical

Of course, that only valid when n is very large, otherwise, it is only what we observe on average….

On the poor performance of classifiers in insurance models

Each time we have a case study in my actuarial courses (with real data), students are surprised to have hard time getting a “good” model, and they are always surprised to have a low AUC, when trying to model the probability to claim a loss, to die, to fraud, etc. And each time, I keep saying, “yes, I know, and that’s what we expect because there a lot of ‘randomness’ in insurance”. To be more specific, I decided to run some simulations, and to compute AUCs to see what’s going on. And because I don’t want to waste time fitting models, we will assume that we have each time a perfect model. So I want to show that the upper bound of the AUC is actually quite low ! So it’s not a modeling issue, it is a fondamental issue in insurance !

By ‘perfect model’ I mean the following : \Omega denotes the heterogeneity factor, because people are different. We would love to get \mathbb{P}[Y=1|\Omega]. Unfortunately, \Omega  is unobservable ! So we use covariates (like the age of the driver of the car in motor insurance, or of the policyholder in life insurance, etc). Thus, we have data (y_i,\boldsymbol{x}_i)‘s and we use them to train a model, in order to approximate \mathbb{P}[Y=1|\boldsymbol{X}]. And then, we check if our model is good (or not) using the ROC curve, obtained from confusion matrices, comparing y_i‘s and \widehat{y}_i‘s where \widehat{y}_i=1 when \mathbb{P}[Y_i=1|\boldsymbol{x}_i] exceeds a given threshold. Here, I will not try to construct models. I will predict \widehat{y}_i=1 each time the true underlying probability \mathbb{P}[Y_i=1|\omega_i] exceeds a threshold ! The point is that it’s possible to claim a loss (y=1) even if the probability is 3% (and most of the time \widehat{y}=0), and to not claim one (y=0) even if the probability is 97% (and most of the time \widehat{y}=1). That’s the idea with randomness, right ?

So, here p(\omega_1),\cdots,p(\omega_n) denote the probabilities to claim a loss, to die, to fraud, etc. There is heterogeneity here, and this heterogenity can be small, or large. Consider the graph below, to illustrate,

In both cases, there is, on average, 25% chance to claim a loss. But on the left, there is more heterogeneity, more dispersion. To illustrate, I used the arrow, which is a classical 90% interval : 90% of the individuals have a probability to claim a loss in that interval. (here 10%-40%), 5% are below 10% (low risk), and 5% are above 40% (high risk). Later on, we will say that we have 25% on average, with a dispersion of 30% (40% minus 10%). On the right, it’s more 25% on average, with a dispersion of of 15%. What I call dispersion is the difference between the 95% and the 5% quantiles.

Consider now some dataset, with Bernoulli variables y, drawn with those probabilities p(\omega). Then, let us assume that we are able to get a perfect model : I do not estimate a model based on some covariates, here, I assume that I know perfectly the probability (which is true, because I did generate those data). More specifically, to generate a vector of probabilities, here I use a Beta distribution with a given mean, and a given variance (to capture the heterogeneity I mentioned above)

a=m*(m*(1-m)/v-1)
b=(1-m)*(m*(1-m)/v-1)
p=rbeta(n,a,b)

from those probabilities, I generate occurences of claims, or deaths,

Y=rbinom(n,size = 1,prob = p)

Then, I compute the AUC of my “perfect” model,

auc.tmp=performance(prediction(p,Y),"auc")

And then, I will generate many samples, to compute the average value of the AUC. And actually, we can do that for many values of the mean and the variance of the Beta distribution. Here is the code

library(ROCR)
n=1000
ns=200
ab_beta = function(m,inter){
  a=uniroot(function(a) qbeta(.95,a,a/m-a)-qbeta(.05,a,a/m-a)-inter,
            interval=c(.0000001,1000000))$root
  b=a/m-a
  return(c(a,b))
}
Sim_AUC_mean_inter=function(m=.5,i=.05){
  V_auc=rep(NA,ns)
  b=-1
  essai = try(ab&lt;-ab_beta(m,i),TRUE) if(inherits(essai,what="try-error")) a=-1 if(!inherits(essai,what="try-error")){ a=ab[1] b=ab[2] } if((a&gt;=0)&amp;(b&gt;=0)){
    for(s in 1:ns){
      p=rbeta(n,a,b)
      Y=rbinom(n,size = 1,prob = p)
      auc.tmp=performance(prediction(p,Y),"auc")
      V_auc[s]=as.numeric(auc.tmp@y.values)}
    L=list(moy_beta=m,
           var_beat=v,
           q05=qbeta(.05,a,b),
           q95=qbeta(.95,a,b),
           moy_AUC=mean(V_auc),
           sd_AUC=sd(V_auc),
           q05_AUC=quantile(V_auc,.05),
           q95_AUC=quantile(V_auc,.95))
    return(L)}
  if((a&lt;0)|(b&lt;0)){return(list(moy_AUC=NA))}}
Vm=seq(.025,.975,by=.025)
Vi=seq(.01,.5,by=.01)
V=outer(X = Vm,Y = Vi, Vectorize(function(x,y) 
Sim_AUC_mean_inter(x,y)$moy_AUC))
library("RColorBrewer")
image(Vm,Vi,V,
      xlab="Probability (Average)",
      ylab="Dispersion (Q95-Q5)",
      col=
        colorRampPalette(brewer.pal(n = 9, name = "YlGn"))(101))
contour(Vm,Vi,V,add=TRUE,lwd=2)

On the x-axis, we have the average probability to claim a loss. Of course, there is a symmetry here. And on the y-axis, we have the dispersion : the lower, the less heterogeneity in the portfolio. For instance, with a 30% chance to claim a loss on average, and 20% dispersion (meaning that in the portfolio, 90% of the insured have between 20% and 40% chance to claim a loss, or 15% and 35% chance), we have on average a 60% AUC. With a perfect model ! So with only a few covariates, having 55% should be great !

My point here is that with a low dispersion, we cannot expect to have a great AUC (again, even with a perfect model). In motor insurance, from my experience, 90% of the insured are between 3% chance and 20% chance to claim a loss ! That’s less than 20% dispersion ! and in that case, even if the (average) probability is rather small, it is very difficult to expect an AUC above 60% or 65% !

Variance decomposition and price segmentation in Insurance

Today, I was giving a talk at the Economics department, and I got a very interesting question about some tables I keep showing to explain why insurance companies like segmentation. The tables illustrate three different case. Here, S stands for the individual (random) loss.

  • the first one is the case where the premium asked is the same for all the insured – i.e. the pure premium \mathbb{E}[S]

As explain, the loss is here on an individual basis, so, per policy, the insurer faces the (random) loss S-\mathbb{E}[S], which is, on average, null. That’s the second line. For the last line, I keep saying that we look at the overall loss of the insurer, but that’s not correct. Here, with a factor n, we would have the variance of the total loss for the insurance company. We just removed the n factor in the table

  • then, we have perfectly observable heterogeneity : insured have a risk factor \Omega, obervable, and in that case, the ‘pure’ premium is \mathbb{E}[S|\Omega]

That’s what we have below. Here again, on average, the insured should have a null profit. And the total variance (which was \text{Var}[S] in our previous example) is now splitted in two parts (that’s basically Pythagoras theorem).

The interpreration is the following

And then, I usually mention the third and last case, more realistic

  • the risk factor \Omega is not obervable, but segmentation is still possible using some proxy of the risk factor, obtained using some covariates, and the ‘pure’ premium is \mathbb{E}[S|\boldsymbol{X}]

And here also, there is a nice interpretation, because of the variance decomposition : there is one part that we observed previously, with some ‘perfect pricing’ and an additional part (that is positive) that is related to the fact that the covariates are just a proxy of the risk factor….

The term on the left is then a lower bound, obtained if actually, using our covariate, available for the pricing, we can get the risk factor.

That was my story, but the fact that n (the portfolio size) was not mentioned in the tables was a bit confusing… So I decided to create some graphs to illustrate those three cases

  • same premium for everyone

Consider some simple simulations. On the graph on the left, we have on the x-axis the risk factor, and on the y-axis, the loss (going roughly from 0 to 20). The pure premium is the average of those losses. Here, it’s 10. That’s the plain red line (on the left). In the middle, the y-axis is the insured profit/loss per policy. Someone with a loss close to 0 means a gain of 10, someone with a loss close to 20 means a loss of 10. On average, there is no profit (that’s the plain line). And then, on the right, we have the distribution of the profit/loss (per contract). Again, on average it’s 0, with some variance,

  • premium based on covariates

Consider here is simple covariate x : assume here that’s we’ve been able to create a binary variable, that can distinguish the low risks and the high risks. Here, there are two levels for the premium. The low premium is close to 6, and the high one is close to 14. That’s again the graph on the left

Then we have the profit/loss per policy for the insured, in the middle. Here, when the loss was close to 0, the gain is smaller : it is 6 (while it was 10 before). When it was close to 10, previously, it meant a 0 profit, but now it’s either a loss of 4, or a gain of 4. The profit/loss distribution is now on the right. There is less dispersion, and less variance. That the decrease of variance we’ve discussed before. To summarize, segmentation does reduce the variability of the result for the insurance company. That’s what we observe on the right.

  • premium based on the risk factor

Assume now that \Omega is observable. And that we use it for our pricing. The premium is now continuous, and it is the red line, on the left. The profit/loss (in the middle) is the difference between the loss, and its expected value (conditional on the risk factor). And on the right, we have the distribution.

As expected, there is much less variability on the profit/loss distribution of the insurance company in that case. And actually, that’s a lower bound for the variance of result of the insurance company… I hope that the graph clarify what’s going on here…

Variance of the slope in a regression model

In my “applied linear models” exam, there was a tricky question (it was a multiple choice, so no details were asked). I was simply asking if the following statement was valid, or not

Consider a linear regression with one single covariate, y=\beta_0+\beta_1x_1+\varepsilon and the least-square estimates. The variance of the slope is \text{Var}[\widehat{\beta}_1] Do we decrease this variance if we add one variable, and consider y=\beta_0+\beta_1x_1+\beta_2x_2+\varepsilon ?

For the exam, the expected answer was simply “no”. In a nutshell, there are two cases where we should expect different changes,

  • if x_1 and x_2 are highly correlated, then we should expect the variance to increase
  • if x_1 and x_2 are not correlated, then we should expect the variance to decrease

We did briefly observed (and discussed) those points on examples during the lecture… but I wanted to go a bit further, since I couldn’t find any analytical results. Let us generate a model y=\beta_0+\beta_1x_1+\beta_2x_2+\varepsilon, and then compare the variance \text{Var}[\widehat{\beta}_1] on the two fitted modes, depending on the correlation between x_1 and x_2

library(mnormt)
n=200
s=function(r=0){
S=matrix(c(1,r,r,1),2,2)
X=rmnorm(n,c(0,0),S)
B=data.frame(y=-2+X[,1]+X[,2]+rnorm(n)/2,
x1=X[,1],
x2=X[,2])
reg12=lm(y~x1+x2,data=B)
reg1=lm(y~x1,data=B)
k=summary(reg12)$coefficients[2,2]/summary(reg1)$coefficients[2,2]
k}

Let us generate 500 samples for each value of the correlation, from -0.9 to _0.9

M=NULL
for(r in ((-9):9)/10) M=cbind(M,Vectorize(s)(rep(r,500)))

and let us plot the ratio of the two variances

plot(0:1,0:1,xlim=c(-1,1),ylim=c(0,2),col="white")
for(i in 1:19) points(rep((((-9):9)/10)[i],500),M[,i],col="light blue")
VM=apply(M,2,mean)
lines((((-9):9)/10),VM,col="red",lwd=2)
abline(h=1,lty=2)

If the ratio exceeds 1, the variance increases when adding a covariate.

Indeed, here, when the two variables are independent, the variance is divided by two. But when covariates are highly correlated, the variance is multiplied by two…

Now, what if, actually, x_2 is not a real explanatory variable : the true model we generate is y=\beta_0+\beta_1x_1+\varepsilon. In that case,

s=function(r=0){
S=matrix(c(1,r,r,1),2,2)
X=rmnorm(n,c(0,0),S)
B=data.frame(y=-2+X[,1]+rnorm(n)/2,
x1=X[,1],
x2=X[,2])
reg12=lm(y~x1+x2,data=B)
reg1=lm(y~x1,data=B)
k=summary(reg12)$coefficients[2,2]/summary(reg1)$coefficients[2,2]
k}

we get our samples as previously

M=NULL
for(r in ((-9):9)/10) M=cbind(M,Vectorize(s)(rep(r,500)))

and we plot those ratios

plot(0:1,0:1,xlim=c(-1,1),ylim=c(0,2),col="white")
for(i in 1:19) points(rep((((-9):9)/10)[i],500),M[,i],col="light blue")
VM=apply(M,2,mean)
lines((((-9):9)/10),VM,col="red",lwd=2)
abline(h=1,lty=2)

In the case we add a useless variable x_2, whatever the correlation with x_1, it will always, on average, increase the variance of \widehat{\beta}_1.

Annual UCSB InsurTech Summit

Just a quick post to mention that an Insurtech Summit will be organized in May 2019, on Friday 3rd, by Mike Ludkovski, and I will be there, with Francois Millard (Vitality Group), Adam Tashman (Carpe Data, Santa Barbara), Emil Valdez (University of Connecticut), and Howard Zail (Elucidor, LLC, New York City). That will be nice… I will actually also give a talk on the Monday before at the actuarial seminar !

An Open Lab-Notebook Experiment