# Strategies and Objectives --- *" You've got to: know when to hold 'em know when to fold 'em know when to walk away know when to run"* -**Kenny Rogers, "The Gambler"** --- ## Goal To discuss how * business objectives * simulations * mathematical analysis can inform and help each other. ---- ## Process To do this, we will consider some simple games that we might play. 1. If you know all the math, please don't ruin it for anyone else. :smiley: 2. If you don't like math, please try anyway. --- ## Situation You play a game with successive turns. 1. On each turn, you have probability $p$ of winning. 2. On each turn, you must invest the same proportion $f$ of your total holdings. 3. If you win, your holdings are matched and returned. 4. You will play for a large number of turns, though you are not sure how many. 5. Goal is to have as much money as possible at the end of the game. --- ## First Situation ### "Double or Nothing" $p = 0.5$ (fair coin) In this game, how much should you invest? You might use [this simulator](https://labs-publications.s3.eu-west-1.amazonaws.com/162/site/index.html?cb=f3878699-e051-4daf-b568-f49f38f43694) to try different parameters. --- ## Optimal Proportion to Invest?                               --- ## Optimal Proportion to Invest?   $0$.                             --- ## Second Situation ### "Double or Nothing" $p = 0.75$ (fair coin) In this game, how much should you invest? You might use [this simulator](https://labs-publications.s3.eu-west-1.amazonaws.com/162/site/index.html?cb=f3878699-e051-4daf-b568-f49f38f43694) to try different parameters and get a feel for the game. --- ## Optimal Proportion to Invest?                             --- ## Optimal Proportion to Invest? $0.5$ (50% of holdings invested on each turn )                             --- ## Beyond Double or Nothing Different return on investment? **Benefit Parameter** $b$: gives return on investment when we win * **Fair Coin Double or Nothing** $b = 1.0$, $p = 0.5$ * **Savings Account with 1% APY** $b = 0.1$, $p = 1.0$ --- ## Changing $b$ and $p$ Discuss the following investment scenarios with your team. What proportion $f$ would you choose? * $p = 0.75$, $b = 0.5$ (75% chance of winning, only get 1.5x investment back) * $p = 0.001$, $b = 99$ (Lottery with small chance of winning, 100x return on investment) You may find the [simulation tool](https://labs-publications.s3.eu-west-1.amazonaws.com/162/site/index.html?cb=f3878699-e051-4daf-b568-f49f38f43694) helpful to gain intuition and vet your own ideas. :) --- ## Kelly Criterion **Result by [John L. Kelly, Jr. (1956)](https://www.princeton.edu/~wbialek/rome/refs/kelly_56.pdf)** $$f^* = p - \displaystyle\frac{1-p}{b}$$ --- ## Kelly Calculations * $p = 0.75$, $b = 0.5$ (75% chance of winning, only get 1.5x investment back) $$f^* = 0.75 - \displaystyle\frac{0.25}{0.5} = 0.25$$ You get the best returns when investing 25% of your funds each round. --- ## Kelly Calculations * $p = 0.001$, $b = 99$ (Lottery with small chance of winning, 100x return on investment) $$f^* = 0.001 - \displaystyle\frac{0.999}{99} $$ **Negative value** stay away from this game. --- ## Connection to Objective Kelly assumed that the goal was to, on average, have the highest value at the **end** of a long game. With a different objective, the strategy may change! --- ## An Alternative Perspective [The viewpoint of Sam Bankman-Fried](https://twitter.com/SBF_FTX/status/1337250686870831107?t=AwEjAOOB0YT2SrrvIIkvMw&s=19) ---