TL;DR I made a web app https://huggingface.co/spaces/mingu600/Tristana_reroll based on 62 million simulations of Tristana reroll where you can input parameters for stage 3-1 and see how much gold you should roll down for 3-1 with the goal of trying to hit all your units level to 6 on Stage 3-5.

Hello, I'm mingu600, and I've been hovering high Master/low GM for most of the set playing Lee Sin Legend pretty much the entire time (https://tactics.tools/player/na/Mingu600), but I've been challenger a few times in previous sets. I play many reroll comps, mainly Tristana and Kayle, and I would say that I have a decent grasp at playing them. However, one thing I've noticed is that not even challenger players actually always agree on the best rolling practices for a comp like Tristana carry.

For example:

• Should you roll down to 30 or 20 on stage 3-1 at level 4, and then econ back to 50, preparing for a roll-down on stage 3-5? This is what I see many challenger flex players do when piloting the comp.
• Should you roll all the way down on stage 3-1, take the Lee Sin 2nd augment to recover gold and snag the last copy with Lesser Duplicator, and aim to winstreak all of stage 3? This is what I see challenger 1-cost one trick Auqaa often doing during his rolldowns.
• Should you slow roll at 50 for multiple rounds, taking advantage of better econ for multiple more rolls?

To my understanding, this isn't actually a solved problem with a definitive answer, and it should depend on the game state. Therefore, I wanted to actually see if there was a difference in success rate between rolldown strategies when piloting Tristana carry using computer simulations. I've seen the previous posts of calculations for average expected rolls to hit X units, but I find that a lot of these are not directly helpful when thinking about actual in-game scenarios in general (how much to roll when? how much gold to hold? when to level? how do augments come into play?). Nothing actually beats playing the game, or at least getting close to it with a simulated version.

Here's how I did it: I coded the shop mechanics, rerolling, buying units, gold income, etc. The computer is trained to pick up copies of Tristana, Maokai, Poppy, and Viego. Whenever it has over 50 gold, it knows how to slow roll down to 50 gold while picking up units. For each simulation, the algorithm gets some number of Tristanas, Maokais, Poppys, and Viegos, as well as a certain amount of gold for stage 3-1. For example, let's say for a particular run we have 5 Tristanas, 4 Maokais, 4 Poppys, and 5 Viegos, and 45 gold at the start of stage 3-1. This is the widely accepted time to aggressively roll down for the Tristana comp. We then give the program a threshold to roll down to on stage 3-1; for example, this particular run might have the program roll down until it reaches 22 gold. I then have the program econ up, only buying relevant units for the comp, until stage 3-5, at which there will be a final rolldown. For some of the simulations, I also added the effects of Trade Sector and Training Reward (averaging silver/gold money amounts) augments.

A success is when we have 3* Tristana, 3* Maokai, 3* Poppy OR Viego, and 24 gold left (to then level to 6 and buy Teemo Jinx) on stage 3-5. Ultimately, we want to hit all of our units with good tempo, and this to me was a reasonable metric for a successful Trist comp rolldown. Each simulation, we go through stages 3-1 to 3-5 over and over, and mark down whether that simulation was a success or a failure, keeping track of the state space as well as the result.

State space: (# of initial Tristanas, # of initial Maokais, # of initial Poppys, # of initial Viegos, whether Trade Sector was taken, whether Training Reward was taken, amount of gold before roll-down, minimum threshold that we rolled down to on 3-1)

Even exploiting symmetries, the state space is very large, which means if we're going to get a large enough sample size for every state, we have to do a LOT of simulations. I ended up doing 62,347,968 total simulations of Tristana reroll!!!

I want to be transparent about some assumptions I make for the simulations:

• Opponents holding units: There is definitely a negative effect if someone is contesting Tristana reroll (or playing Kayle reroll). In a real scenario however, you two are going 7th/8th, so we will just safely assume no one is contesting the comp. Thus, because this is a 1-cost reroll, the effect of opponents holding non-Tristana related units is somewhat minimal, and due to the rise of other 1-cost reroll comps, hopefully ends up being balanced out. The simulations do take into account the units that we are holding however.
• Bench space: We assume bench space is unlimited. Realistically this can be a concern sometimes in the worst case scenarios- in practice, just note that people tend to lock their shop, play a new unit on stage 3-2 once you reach level 5 and then your bench frees up a bit.
• Streaking: We assume that you have successfully loss-streaked all of stage 2. On stage 3 however, I made up (out of my ass) how to handle chances of winning a round. If anything, these winrates might be too high, but it's basically used for the sole purpose of calculating streak contribution of interest gold.
  • 4* Tristana: 100% win
  • 3* Tristana 3* Maokai on board: 50% win
  • Either 3*Tristana or 3*Maokai: 30% win
  • Otherwise: 10% win
• Effects of HP: Ideally, a true objective function will also include how healthy you currently are. For example, it might be better to roll for a stronger board early in order to save one more life by winning a round. This leads to a whole line of even more assumptions though based on opponent boards that we aren't simulating, so I decided to leave HP considerations out of the scenario.
• Effects of portals: Money gained from Ecliptic Vaults, champion duplicators gained from The Sump, etc. are not considered. Other portals such as Scuttle Puddle can be accounted for by simply giving the resulting initial gold on 3-1.

Here's the code https://github.com/mingu600/Tristana-Reroll (it's quite messy with lots of commented out code, sorry, and it might not run as is since I've made touches here and there but hopefully you can see the logic). Please let me know if you see something wrong in the actual logic! I have tried to look carefully and print out every step, but there's definitely things I could have missed.

The App

I deployed an app here (https://huggingface.co/spaces/mingu600/Tristana_reroll) using Gradio through HuggingFaces based on the results. I know nothing about app dev, web dev, etc. so I just used a tool that I've used before for ML. Here's a screenshot:

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Tristana Reroll app

On the left, you can change the inputs depending on whether you have Trade Sector, whether you plan on taking Training Reward on Stage 3-2, the number of Tristanas, Maokai, Poppy, and Viego you have, and the amount of starting gold (limited to 40, 45, and 50). After pressing Submit, it should generate two plots. On the top right is a counts histogram of all the trials with your exact initial inputs, and all the results based on how far the program rolled down on Stage 3-1, reds being failures and blues being successes. The bottom right plot shows the probability of success based on the simulations for each amount of gold rolled down to. For this specific scenario, I would want to roll down to 32 gold; the winrate does not increase beyond, and if so then I would want a higher chance of an earlier stronger board in order to save HP compared to saving even more gold.

In general, I've seen that rolling only down to 30 is ultimately better than rolling more aggressively in terms of satisfying the objective of reaching level 6 with all our units on Stage 3-5.

Please let me know if you have any questions or comments!