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Miscellaneous

Here I present some miscellaneous work/blog posts on various topics that aren't necessarily related to any of my research.

If you have any questions or comments regarding any of my work, please feel free to email me.

Risk, Uncertainty, and Ambiguity

  1. An Introduction to the Concepts of Risk, Uncertainty, and Ambiguity

    Abstract: A very short introduction to the concepts of risk, uncertainty, and ambiguity--and their differences--written from the perspective of an economist.

Virginia Online Lottery

  1. An Introduction to How the Online Lottery Works

    Abstract: This is the first part of a blog post originally posted to Reddit to help users of the VA Online Lottery better understand how the online games work and the odds involved. It discusses the distribution of payoffs and how a particular payoff is randomly selected for any given play.
  2. Are the Virginia Online Lottery Promotions Rigged? An Analysis of Expected Earnings Using Simulation Methods

    Abstract: I use simulation methods to better understand the distribution of earnings that result from playing a series of gambles in an Online Instant game (i.e., a virtual slot machine) on the Virginia Lottery website. The VA Online Lottery frequently offers players promotions that shift the odds in the player's favor, such as providing players with a 100% match on deposits made to their account. (This effectively makes the online games half-price, and shifts the odds so that on average players will earn positive profits.) However, despite the expected profits associated with these promotional offers, many players who respond to the incentives find that they actually end up losing money, many times finding that they win almost nothing. This has resulted in many players suspecting that the Virginia Online Lottery is rigged. To investigate the merits of these suspicions, I built a simulation model of one of the Online Instant games offered on the Virginia Online Lottery website and simulated various promotional offers to see how often one should expect them to pay a positive profit. I presented my results to online gamblers to help them better understand their odds of winning (and losing).
  3. Do Deposit Matches Ever Guarantee a Profit? When is a Deal Too Good to Pass Up?

    Abstract: Using a simulation model of the Virginia Online Lottery game Jackpot Spectacular, I analyze the expected profits associated with various deposit match promotions offered by the Lottery. By simulating the distributions of expected profits for each promotion, I am able to determine how often players should expect to profit and answer the question, "When is a deal too good to pass up?"
  4. Is There an Optimal Bet Size for Winning a Progressive Jackpot?

    Abstract: Unlike regular prizes in Online Instant games, the odds of winning a progressive jackpot scale with the size of the bet placed. Because of this, players can increase the odds of winning the jackpot in any given play by increasing the size of the bet placed. However, for a player with a fixed budget, increasing the bet size reduces the number of games they can play, giving them fewer opportunities to win the jackpot. So which strategy is optimal, assuming the player wants to maximize their odds of winning the jackpot? I answer this question and explain why theory says players should place the largest bets possible.
  5. A More Mathematically Rigorous Analysis of the Optimal Bet Size

    Abstract: I provide a more mathematically rigorous model of a lottery with a progressive jackpot and use the model to determine the optimal strategy for a player seeking to maximize their chances of winning the jackpot.
  6. A More Realistic Analysis of the Optimal Bet Size, with Simulation

    Abstract: I update my analysis to be more realistic by allowing players to use their winnings to continue playing. Rather than playing a fixed number of games, the total number of games played will depend on the payoffs the player receives. Furthermore, if a player's budget drops below their desired bet size but they still have enough funds to play at some lower bet size, they move to the largest bet size they can afford and continue playing. All players continue playing until they either win the jackpot or they fully exhaust their funds (to the point that they cannot afford the minimum bet size). These additions to the framework make the analysis much more realistic, but they also greatly increase the complexity. To get an understanding of what bet size is optimal within this more realistic framework, I build a simulation model of the actual online game Jackpot Spectacular, and analyze the probability of winning the jackpot using a range of bet sizes. Unlike my previous work, I now find that smaller bet sizes provide a significantly higher chance of winning the jackpot.

"The invariable mark of wisdom is to see the miraculous in the common."

---Henry David Thoreau