The Kelly criterion for spread bets

The optimal betting strategy for a gambler betting on a discrete number of outcomes was determined by Kelly (1956, A new interpretation of information rate. J. Oper. Res. Soc., 57, 975–985). Here, the corresponding problem is examined for spread betting, which may be considered to have a continuous distribution of possible outcomes. Since the formulae for individual events are complicated, the asymptotic limit in which the gamblers edge is small is examined, which results in universal formulae for the optimal fraction of the bank to wager, the probability of bankruptcy and the distribution function of the gamblers total capital

Elicitation of ambiguous beliefs with mixing bets

I consider the elicitation of ambiguous beliefs about an event and show how to identify the interval of relevant probabilities (representing ambiguity perception) for several classes of ambiguity averse preferences. The agent reveals her preference for mixing binarized bets on the uncertain event and its complement under varying betting odds. Under ambiguity aversion, mixing is informative about the interval of beliefs. In particular, the mechanism allows to distinguish ambiguous beliefs from point beliefs, and identifies the belief interval for maxmin preferences. For ambiguity averse smooth second order and variational preferences, the mechanism reveals inner bounds for the belief interval, which are sharp under additional assumptions. In an experimental study, participants perceive almost as much ambiguity for natural events (generated by the stock exchange and by a prisoners dilemma game) as for the Ellsberg Urn, indicating that ambiguity may play a role in real-world decision making

Kelly Betting Can Be Too Conservative

Kelly betting is a prescription for optimal resource allocation among a set of gambles which are typically repeated in an independent and identically distributed manner. In this setting, there is a large body of literature which includes arguments that the theory often leads to bets which are "too aggressive" with respect to various risk metrics. To remedy this problem, many papers include prescriptions for scaling down the bet size. Such schemes are referred to as Fractional Kelly Betting. In this paper, we take the opposite tack. That is, we show that in many cases, the theoretical Kelly-based results may lead to bets which are "too conservative" rather than too aggressive. To make this argument, we consider a random vector X with its assumed probability distribution and draw m samples to obtain an empirically-derived counterpart Xhat. Subsequently, we derive and compare the resulting Kelly bets for both X and Xhat with consideration of sample size m as part of the analysis. This leads to identification of many cases which have the following salient feature: The resulting bet size using the true theoretical distribution for X is much smaller than that for Xhat. If instead the bet is based on empirical data, "golden" opportunities are identified which are essentially rejected when the purely theoretical model is used. To formalize these ideas, we provide a result which we call the Restricted Betting Theorem. An extreme case of the theorem is obtained when X has unbounded support. In this situation, using X, the Kelly theory can lead to no betting at all.Comment: Accepted in 2016 IEEE 55th Conference on Decision and Control (CDC

Measuring Belief and Risk Attitude

Ramsey (1926) sketches a proposal for measuring the subjective probabilities of an agent by their observable preferences, assuming that the agent is an expected utility maximizer. I show how to extend the spirit of Ramsey's method to a strictly wider class of agents: risk-weighted expected utility maximizers (Buchak 2013). In particular, I show how we can measure the risk attitudes of an agent by their observable preferences, assuming that the agent is a risk-weighted expected utility maximizer. Further, we can leverage this method to measure the subjective probabilities of a risk-weighted expected utility maximizer

Universal statistical properties of poker tournaments

We present a simple model of Texas hold'em poker tournaments which retains the two main aspects of the game: i. the minimal bet grows exponentially with time; ii. players have a finite probability to bet all their money. The distribution of the fortunes of players not yet eliminated is found to be independent of time during most of the tournament, and reproduces accurately data obtained from Internet tournaments and world championship events. This model also makes the connection between poker and the persistence problem widely studied in physics, as well as some recent physical models of biological evolution, and extreme value statistics.Comment: Final longer version including data from Internet and WPT tournament

Interpreting the Predictions of Prediction Markets

Participants in prediction markets such as the Iowa Electronic Markets trade all-or-nothing contracts that pay a dollar if and only if specified future events occur. Researchers engaged in empirical study of prediction markets have argued broadly that equilibrium prices of the contracts traded are market probabilities' that the specified events will occur. This paper shows that if traders are risk-neutral price takers with heterogenous beliefs, the price of a contract in a prediction market reveals nothing about the dispersion of traders' beliefs and partially identifies the central tendency of beliefs. Most persons have beliefs higher than price when price is above 0.5, and most have beliefs lower than price when price is below 0.5. The mean belief of traders lies in an interval whose midpoint is the equilibrium price. These findings persist even if traders use price data to revise their beliefs in plausible ways.

Decision-Making in the Context of Imprecise Probabilistic Beliefs

Coherent imprecise probabilistic beliefs are modelled as incomplete comparative likelihood relations admitting a multiple-prior representation. Under a structural assumption of Equidivisibility, we provide an axiomatization of such relations and show uniqueness of the representation. In the second part of the paper, we formulate a behaviorally general axiom relating preferences and probabilistic beliefs which implies that preferences over unambiguous acts are probabilistically sophisticated and which entails representability of preferences over Savage acts in an Anscombe-Aumann-style framework. The motivation for an explicit and separate axiomatization of beliefs for the study of decision-making under ambiguity is discussed in some detail.

Inter-market Arbitrage in Sports Betting

Unlike the existing literature on sports betting, which concentrates on arbitrage within a single market, this paper examines inter-market arbitrage by searching for arbitrage opportunities through combining bets at the bookmaker and the exchange market. Using the posted odds of eight different bookmakers and the corresponding odds traded at a well-known bet exchange for 5,478 football matches played in the top-five European leagues during three seasons, we find (only) ten intra-market arbitrage opportunities. However, we find 1,450 cases in which a combined bet at the bookmaker as well as at the exchange yields a guaranteed positive return. Further analyses reveal that inter-market arbitrage emerges from different levels of informational efficiency between the two markets.sports betting, inter-market arbitrage