Trader Brain Teasers | Bet-Sizing | Interview Questions | Brackets/Horse Game

How do you approach these questions? I believe the entire point of asking these questions is to gauge your knowledge of Kelly Criterion or knowing how to optimally size bets given you know the odds and probabilities of success. However, I am unaware of how to size my bets in a horse game/bracket betting situation. (These two situations are typically in Optiver/Market-Maker questions/brain teasers and I'm hoping to prepare because I can't find definitive solutions for these, or very few questions to even test on.)


I'll post the two situations with arbitrary numbers and explain what I believe the answer would be, but I'd appreciate someone much smarter than I to verify this thought process.


Situation 1: Horse Racing. There are four horses ABCD, A has a payout of 2/1 and has a 33% chance of winning, B has a 1/2 payout and has a 17% chance of winning, C has a 4/1 payout and has a 40% chance of winning, and D has an 10/1 payout and has a 10% chance of winning. Given you have $100, how much would you bet on each horse and why - if you would not bet on certain horses explain why.

My thought process: 1st I want to break this down into expected value terms and take note of the horse that has the highest POSITIVE expected value in terms of return based on chance of winning. After I realize which one has the highest positive EV, that is the single horse I will be placing my bet on because in my opinion, it doesn't make sense to bet on another horse even if it has positive EV because I can get an even higher return betting on the horse that has a higher EV. In this case, for every dollar you bet on horse A, you receive 2 back if you win and lose 1 if you lose. EV horse A calculation +2.0 * 0.33, -1 * 0.66. EV is 0. EV horse B would be negative right off the bat, 83% to lose 2, 17% chance to gain 1. I don't even want to look at this horse now. Horse C EV is the greatest. +4 * 0.4, -1 * 0.6. A positive EV of + 1. Horse D EV is +10 * 0.1, -1 * 0.9. It still has positive EV of +0.1, but I don't think I should consider this horse at all because Horse C has the highest EV.

Now that I know which horse has the highest EV and the probability of winning, I will calculate my bet sizing for this horse specifically. I calculate the optimal bet for this horse alone using its odds and get: 0.4 - (0.6/(b)) where b = 4/1) = 25%.

My Answer: I will bet 25% of my money on horse C, 0 on horse D, 0 on horse B, 0 on horse A. Does that sound like the proper thought process, or is there a way to split the money up further? If I start taking money out of the betting pool to put on other horses, I lower my money bet but I could potentially increase the amount of money throughout the pot. If there's -EV at all, I will not bet, but for any positive EV situations since it's a % of my bank roll I could divy it up more? Bet 25% on horse C, then look at horse D 0.10-(.9/10) = 1% on horse D, so $25 horse C, 0.01 * 75 (remaining money after horse C) for $0.75 horse D



Situation 2: Semi-Finals Bracket Betting Scenario between 4 teams ABCD.

Semi Finals

Scenario 1: A vs C | 50/50 odds (1/1)

Scenario 2: B vs D | 30/70 odds (2.33/1)

Finals

Scenario 3: A vs B | If A wins semi finals and B wins semifinals, new odds are 10/90 (9/1) for team A

Scenario 4: A vs D | If A wins semi finals and D wins semifinals, new odds are 60/40 (.67/1) for team A

Scenario 5: C vs B | If C wins semifinals and B wins semifinals, new odds are 30/70  (2.33/1) for C

Scenario 6: C vs D | If C wins semifinals and D wins semifinals, new odds are 60/40 (.67/1) for team D


What % of your capital would you bet during each round to maximize your profit? (Additionally I forgot if there was a $ reward, there probably was so I'm more so kind of just looking for how you would break this problem down. Looks like TLDR the concept is | 


Approach: Honestly, I don't really know how to approach this and I don't know if I even structured the problem correctly. I just remember a specific bracket-betting problem on Optiver's math test and was trying to remember it from 5 months ago. I don't know if these numbers even make sense for what I'm trying to ask, but basically I am wondering if I was given a bracket betting question in terms of knowing the probabilities each team has to win based on what happens, how much should I bet on the semi-finals and the final rounds? In scenario 1 it's 50/50, and in scenario 2 it's more likely that D move on to the finals. But now if D moves on to the finals, they have worse odds against A or C. So even though D was favored to move on to the finals, they are not the most favored to win it. If anyone remembers this problem or know something similar to it, please feel free to chime in.

 

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