In-game betting and the Kelly criterion

When a bet with a positive expected return is available, the Kelly crite-rion can be used to determine the fraction of wealth to wager so as to maximizethe expected logarithmic return on investment. Several variants of the Kelly cri-terion have been developed and used by investors and bettors to maximize theirperformance in inefficient markets. This paper addresses a situation that has not,hitherto, been discussed in academic literature: when multiple bets can be placedon the same object and the available odds, true probabilities, or both, vary overtime. Such objects are frequently available in sports betting markets, for example,in the case of in-game betting on outcomes of soccer matches. We adapt the Kellycriterion to support decisions in such live betting scenarios, and provide numericalexamples of how optimal bet sizes can sometimes be counter-intuitive

Rate-Optimal Multiuser Scheduling with Reduced Feedback Load and Analysis of Delay Effects

In this paper we propose a multiuser scheduling algorithm that has the maximum average system spectral efficiency, but obtains a significant reduction in feedback load compared to full feedback by using a feedback threshold. An expression for the threshold value that minimizes the feedback load is found. Novel closed-form expressions are also found for the system spectral efficiency when using M-ary quadrature amplitude modulation. Finally, we analyze the impact of scheduling delay and outdated channel estimates. 1

Design Issues and Performance Analysis for Opportunitistic Scheduling Algorithms in Wireless Networks

This doctoral thesis is a collection of six papers preceeded by an introduction. All the papers are related to design issues and performance analysis for opportunistic scheduling algorithms in cellular networks. Opportunistic scheduling algorithms can provide higher throughput and increased quality-of-service (QoS) in wireless networks by giving priority to the users with favorable channel conditions. Such algorithms are already implemented in equipment based on wireless LAN standards, the HDR standard, the HSDPA standard, and the Mobile WiMAX standard, but are often not a part of the standard itself. The implemented algorithms are often based on intuition rather than theoretical investigations, and consequently, it is a need for a better understanding of the theoretical limits for how well such algorithms can perform and how such algorithms can be implemented in the most efficient way. The design issues handled in this thesis are related to feedback algorithms and scheduling algorithms for increased throughput guarantees. Channel quality estimation and feedback is the basis for opportunistic scheduling, and two novel feedback algorithms are proposed to reduce the overhead from channel quality feedback. The results show that the feedback can be reduced to only obtaining feedback from the user that the system wants to schedule. An adaptive scheduling algorithm for obtaining increased throughput guarantees is also developed. Results from simulations show that this algorithm can double the throughput guarantees in modern cellular networks compared to other well-known scheduling algorithms. The performance of opportunistic scheduling algorithms is analyzed through analytical expressions and simulations of feedback delay, fairness and throughput guarantees. It is shown how delayed feedback can lead to reduced throughput or increased bit error rate in a system with opportunistic scheduling. Closed-form expressions are also found for two types of fairness, and throughput guarantees of different well-known scheduling algorithms

In-game Betting and the FA English Premier League: The Contribution of Prediction Models

Den samlede verdien på markedet for live-betting på kamper i FA English Premier League øker raskt. For å kunne tjene penger i et slikt marked trenger man, som i alle finansielle markeder, pålitelig informasjon on fremtidige hendelser og gode investeringsstrategier. Denne oppgaven er skrevet i samarbeid med Sportradar AG, som tilbyr prediksjonstjenester til tilbydere av odds i sportsmarkeder. Med et mål om å forbedre disse prediksjonene blir et sett av prediksjonmodeller generert og evaluert. Modeller av den bivariate scoringsdistribusjonen og HUB-distribusjonen er generert fra to ulike arkitekturer. Den første arkitekturen er basert på et long short-term memory-nettverk og den andre bygger på Weibull count-distribusjonen (McShane et al., 2008). I denne arkitekturen brukes Frank copula for å modellere avhengigheten mellom scoringsprosessene til to motstandere. Alle modellene er trent ved å minimere cross entropy, som er sammenfallende med maximum likelihood-estimering. Sammenligninger mellom modellene er gjort for å avgjøre om det er best å benytte en parametrisk distribusjon eller en kompleks black-box-metode. Dette er motivert av en hypotese om at det første alternativet medfører unødvendige restriktjoner på læringsprosessen. I tillegg sammenlignes ytelsen til scoringsmodellene og HUB-modellene for å teste om informasjon om scoringsdistribusjonen er essensiell for å modellere HUB-distribusjonen nøyaktig. Resultatene tyder på at alle modellene har nesten lik prediksjonsevne basert på både cross entropy og accuracy score når de måles på HUB-distribusjonen. De beste resultatene som observeres er henholdsvis 0.9423 og 0.5783. Modellene som er basert på Weibull count-distribusjonen har gjennom hele kampen generelt litt bedre ytelse enn de andre modellene, når ytelsen måles ut fra cross entropy og accuracy score. Ranked probability score sier at long short-term memory-modellene er signifikant best hvis man tar hensyn til den ordinale strukturen i scoringsdistribusjonen. Betting-resultatene for de ulike modellene er basert på gode, teoretiske investeringsstrategier. En av disse er en dynamisk Kelly-strategi foreslått av forfatterene. Den andre er en strategi som brukes for å teste den statiske ytelsen til modellene og for å være en referanse for den dynamiske strategien. De statiske resultatene indikerer at alle prediksjonsmodellene genererer positiv avkastning i enkelte tidspunkt av kampene ved bruk av partial-Kelly strategier. Den beste modellen kombinert med en risiko-avers partial-Kelly strategi genererer ofte avkastning opp mot 15 % og indikerer dermed at potensialet i prediksjonene er godt. Den dynamiske strategien gir høyere og mer volatil avkastning enn den statiske. Ved å bruke en partiell Kelly-parameter på 0.05 og sannsynligheter fra en av Weibull distribusjons-modellene ga den 30 % avkastning med en akseptabelt lav volatilitet. Det er likevel ingen kombinasjon av prediksjonmodell og strategi som konsekvent gir positiv avkastning

Feedback Protocols for Increased Multiuser Diversity Gain in Cellular ALOHA Networks : A Comparative Study

Multiuser diversity (MUD) underlies much of the recent work on scheduling design in wireless networks. This form of diversity can for example be exploited by opportunistically scheduling the mobile user with the best channel quality. In cellular networks exploiting MUD, the base station collects channel state information (CSI) from the mobile users. The process of obtaining CSI will be performed within a guard time, and the length of this guard time will depend on the feedback protocol implemented. In this context, it has already been shown that by applying multiple carrier-to-noise ratio thresholds, the number of mobile users giving feedback can be significantly decreased. However, it has not been evaluated how the algorithm in can be implemented in protocols for real-life networks. In this paper we analyze feedback protocols for reducing the guard time and resolving the feedback contention problem in a cellular, slotted ALOHA-based network. We propose three new feedback protocols based on the algorithm in and we develop closed-form expressions for the guard time duration and the system spectral efficiency of these protocols. We also compare the three new protocols with the Splitting algorithm proposed by Qin and Berry and a new and modified version of this algorithm. Plots show that the spectral efficiency in an IEEE 802.11 network can increase significantly for a high number of users when the Modified Splitting algorithm is used