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The Math Behind Surrendering

Subdue··3 votes·906 views

Assuming: Avg LP Gained Per Win = 20 LP Avg LP Lost Per Loss = 20 LP Avg Game Length 30 Minutes

The expected value per minute of surrendering at 20 minutes is -1 LP / Min.

According to www.leagueofgraphs.com, the probability of winning after losing key objectives is as follows:

First Inhibitor 8% First Baron 18% First Dragon 31% First Tower 27% First Blood 40%

Therefore, if you play out a game you were expecting to lose at 20 minutes to 30 minutes, the average game length, the expected value per minute of the game becomes:

First Inhibitor -0.56 LP / Min First Baron -0.43 LP / Min First Dragon -0.25 LP / Min First Tower -0.30 LP / Min First Blood -0.13 LP / Min

In other words, even in the worst statistic, losing your Inhibitor first, and having only an 8% likelihood of winning, not surrendering improves the value of your LP / Min by 44%.

Remember this the next time you hit that surrender button at 20 minutes.

Edit: Updated values - They were even better than I thought. Thank you to No to Toxic for catching my math error. :P

31 Comments

I Eat Ąss3 votes

You can't just correlate random things with statistics and say they apply

Subdue3 votes

[{quoted}](name=I Eat Ąss,realm=NA,application-id=3ErqAdtq,discussion-id=L1i6IjMu,comment-id=0000,timestamp=2018-03-13T19:09:59.547+0000)

You can't just correlate random things with statistics and say they apply

Way to object without actually objecting to anything specific. Makes for a solid argument.

I Eat Ąss3 votes

I honestly don't even know where to begin.

Game Length =/= Value of LP.

In a game with an 8 percent likelihood that you win and you don't surrender at 20 min, every additional minute is worse for your LP because you could be in another game Winning LP.

A result that is certain does not decrease the penalty value if the penalty is a stagnant. You lose 20 LP no matter what and if you have an 8 percent chance of winning then you have a 92 percent chance of losing, making -20 the more statistical probability.

Take into account that that value will only get worse due to snowballing and the enemy getting more objectives and you are actually in a worse position by not surrendering a near certain loss and the Value of your LP or LP Gain/Loss has actually decreased because you lock yourself in to a 20 LP loss instead of spending the 20 extra minutes you played breaking even in another game

Subdue2 votes

[{quoted}](name=I Eat Ąss,realm=NA,application-id=3ErqAdtq,discussion-id=L1i6IjMu,comment-id=000000000000,timestamp=2018-03-13T19:19:39.147+0000)

I honestly don't even know where to begin.

Game Length =/= Value of LP.

In a game with an 8 percent likelihood that you win and you don't surrender at 20 min, every additional minute is worse for your LP because you could be in another game Winning LP.

A result that is certain does not decrease the penalty value if the penalty is a stagnant. You lose 20 LP no matter what and if you have an 8 percent chance of winning then you have a 92 percent chance of losing, making -20 the more statistical probably.

Take into account that that value will only get worse due to snowballing and the enemy getting more objectives and you are actually in a worse position by not surrendering a near certain loss and the Value of your LP or LP Gain/Loss has actually decreased because you lock yourself in to a 20 LP loss instead of spending the 20 extra minutes you played breaking even in another game

  1. Game Length is totally related to the value of your LP. After all, it's the time you're trying to save by surrendering isn't it? Putting it another way: If a win is worth 50 LP and a loss is worth -1, you'd surrender any time a game looked like it was even a little bit disadvantageous, because the expected value of any game before it starts is 24.5. Using the same logic, if you know the length of the game, and the result of a win or a loss, you can calculate the value of each minute of the game. Since most people gain per win about as much as they lose per loss, there's no point in factoring in the value of the next game, because the next game has an expected value of 0.

  2. Very rarely is a result "certain" at 20 minutes. Even if the enemy has taken an inhibitor by then, statistics show that teams still pull out a win 8% of the time. Therefore, while the penalty for the loss does not go up, by not surrendering, you extend the possibility of the win.

  3. The value of snowballing is already factored into the probabilities. The First Tower statistic of a 40% chance to win after losing first tower already encompasses all of the possible results of losing the first tower, and how that may or may not snowball a game to a loss. And as I said, unless you are gaining massively more LP per win than you're losing per loss, there's no point factoring in future games, as the expected value is 0.

I Eat Ąss2 votes

This is a very crude way to understand and use statistics but i guess?

Also, 85% of games are pretty much decided at 20 minutes. I don't know what you are talking about with "rarely"

That was a good joke though

Subdue2 votes

[{quoted}](name=I Eat Ąss,realm=NA,application-id=3ErqAdtq,discussion-id=L1i6IjMu,comment-id=00000000000000000000,timestamp=2018-03-13T19:39:49.701+0000)

This is a very crude way to understand and use statistics but i guess?

Also, 85% of games are pretty much decided at 20 minutes. I don't know what you are talking about with "rarely"

That was a good joke though

93.8% of statistics are made up on the spot.

Azadethe1 votes

Game length can respresent the value of LP, because you are calculating how a surrender is valued, and the only common factor we evaluate in a surrender vs a prolonged game is "time wasted vs time saved. Time is the common variable.

no to toxic1 votes

Can you atleast show the math of your statistics as it seems that the lp loss increase is a bit too low.

Subdue1 votes

[{quoted}](name=no to toxic,realm=EUW,application-id=3ErqAdtq,discussion-id=L1i6IjMu,comment-id=0001,timestamp=2018-03-13T19:28:11.096+0000)

Can you atleast show the math of your statistics as it seems that the lp loss increase is a bit too low.

Sure!

So, for a surrender at 20, you will take the LP loss 100% of the time. So if your LP loss is 20, then that's 20 LP / 20 Mins, or 1 LP / Min.

I'll use lost First Tower as an example for the others and you can repeat the math. If you choose to play out the game to the average length of 30 minutes, then your expected value of LP / Min is as follows.

(Chance to Win)x(LP For Win)/(Game Length) + (Chance to Lose)x(LP for Loss)/(Game Length) (.40)x(20)/(30) + (.60)x(-20)/(30) = -.01333 LP / Min

redniwediS1 votes

In a vacuum 10 additional minutes may increase my LP/Min, but those 10 minutes could also lower the likelyhood of victory in subsequent games due to tilt. There is a human part to this equation that your numbers do not take into account.

Subdue1 votes

{quoted}

In a vacuum 10 additional minutes may increase my LP/Min, but those 10 minutes could also lower the likelyhood of victory in subsequent games due to tilt. There is a human part to this equation that your numbers do not take into account.

You should know you're tilted and not play the next game, or play a norm, or an aram or something.

redniwediS1 votes

That is what I do, but the more I'm tilted the more of a break I have to take, and so the less LP/Min I get. Those 10 minutes can cancel out an afternoon of effective play time and lower my LP gain significantly, to the point where the victory of that particular game is nowhere near worth my time.

Xion The XIV1 votes

So what happens when you lose all of those things? I now have 376% chance of losing So that questions if all of that happens I should not surrender, because someone believes there is a chance of winning? But again how can these statistics be correct with the game like LoL? Does it take in account DCers/Trollers/Inters? These 3 type of people are in the average of games, so if they didn't take into account of these type of people how can they be accurate?

Subdue1 votes

{quoted}

So what happens when you lose all of those things? I now have 376% chance of losing So that questions if all of that happens I should not surrender, because someone believes there is a chance of winning? But again how can these statistics be correct with the game like LoL? Does it take in account DCers/Trollers/Inters? These 3 type of people are in the average of games, so if they didn't take into account of these type of people how can they be accurate?

That's not how that works. You don't add those percentages together. Built into the 40% for the tower is already the X% of the time that the tower snowballed into an inhibitor, for example. You take the worst one that applies to your game.

Azadethe1 votes

The stats are not mutually exclusive so you wouldn't apply them additively. You'd apply them cumulatively (1 + u)(1 + u) etc ....... -1

In addition, there is also a flaw with her argument. The expected value of a loss is -2 LP/Minute, not -1, because you have to account for both the value of the current game, and the value of the previous game (not just one game) because a loss puts the value of a current win @ 0 while also putting the value of the PREVIOUS game at - 20.

haaaaaaalp1 votes

Where do you get the idea that the expected value of surrendering is -1 lp per minute?

Your expected lp per minute is based on the lp you would expect to lose from surrendering minus the lp you would expect to gain from starting your next game early.

If you are below the elo you belong, your expected lp from starting a new game will usually end up being higher than your expected loss lp from ffing. If you are at the lp you belong, your expected lp over time will be zero reguardless of whether or not you ff. Ffing will only decrease your expected lp loss over time if you are above the elo you belong.

Subdue1 votes

{quoted}

Where do you get the idea that the expected value of surrendering is -1 lp per minute?

Your expected lp per minute is based on the lp you would expect to lose from surrendering minus the lp you would expect to gain from starting your next game early.

If you are below the elo you belong, your expected lp from starting a new game will usually end up being higher than your expected loss lp from ffing. If you are at the lp you belong, your expected lp over time will be zero reguardless of whether or not you ff. Ffing will only decrease your expected lp loss over time if you are above the elo you belong.

That's a fair critique. So a bit of additional math!

Let's say that your MMR is higher than your Tier/Division show, so you're earning 25 LP per win, and losing 15 LP per loss.

The Expected Value of your games then is 10, or over a 30 minute game, .33333 LP / Min.

Thus, the penalty for surrendering is -15 LP + .33333 LP / Min x 10 Min, or -11.6667 LP. Spread this across the full 30 minutes, and you end up with -.38889 LP / Min, which is just about equal with staying after you've lost your first inhibitor, and worse than staying under all the lesser predictors.

If you're super underrepresented by your Tier/Division because your MMR is that much higher, and you're earning 30 per win and losing 10 per loss or something like that, then it might be better to cut your losses early and move on to the next game. Of course, that lowers your MMR, reducing the effectiveness of subsequent wins, though that's more complicated math than I'd like to get into. For most players, you are likely right at the sweet spot with even gains or losses, or at most are in the above scenario, earning 25 losing 15.

haaaaaaalp1 votes

{quoted}

That's a fair critique. So a bit of additional math!

Let's say that your MMR is higher than your Tier/Division show, so you're earning 25 LP per win, and losing 15 LP per loss.

The Expected Value of your games then is 10, or over a 30 minute game, .33333 LP / Min.

Thus, the penalty for surrendering is -15 LP + .33333 LP / Min x 10 Min, or -11.6667 LP. Spread this across the full 30 minutes, and you end up with -.38889 LP / Min, which is just about equal with staying after you've lost your first inhibitor, and worse than staying under all the lesser predictors.

If you're super underrepresented by your Tier/Division because your MMR is that much higher, and you're earning 30 per win and losing 10 per loss or something like that, then it might be better to cut your losses early and move on to the next game. Of course, that lowers your MMR, reducing the effectiveness of subsequent wins, though that's more complicated math than I'd like to get into. For most players, you are likely right at the sweet spot with even gains or losses, or at most are in the above scenario, earning 25 losing 15.

Lowing your mmr just makes the games much easier to win. Your expect lp per minute should be much higher at that point. You are trading an 8% to turn 40 for a 60% chance to turn 10

Subdue1 votes

{quoted}

Lowing your mmr just makes the games much easier to win. Your expect lp per minute should be much higher at that point. You are trading an 8% to turn 40 for a 60% chance to turn 10

Lowering your MMR makes the next game easier, but if you win it, you're basically back where you were before in terms of difficulty. If your intent is to climb the ladder, losing MMR only hurts you.

haaaaaaalp1 votes

Sitting through a 20 minute lost game will hurt your mmr. FFing and winning the next one puts you back where you start

Subdue1 votes

{quoted}

Sitting through a 20 minute lost game will hurt your mmr. FFing and winning the next one puts you back where you start

And FFing and losing again puts you twice as far behind... What's your point?

haaaaaaalp1 votes

{quoted}

And FFing and losing again puts you twice as far behind... What's your point?

Mathematically, you are more likely to come out ahead by ffing your lost games when you are below your true elo and even if you are at your true elo. No reason to drag out lost games unless you feel you are above your true elo.

Overall, the elo model will get you to where you belong reguardless of whether or not you ff your lost games. FFing just changes the speed at which you get there.

Subdue1 votes

{quoted}

Mathematically, you are more likely to come out ahead by ffing your lost games when you are below your true elo and even if you are at your true elo. No reason to drag out lost games unless you feel you are above your true elo.

Overall, the elo model will get you to where you belong reguardless of whether or not you ff your lost games. FFing just changes the speed at which you get there.

FFing only speeds things up if your MMR is way higher than the average MMR of your Tier/Division, and even then, only in the worst of scenarios, as the math clearly shows. When your MMR matches the average of your Tier/Division you are better off not surrendering, and when it's way below, you actually hurt yourself by surrendering.

eaglei31 votes

How are you getting a 44% improvement for first inhib?

Subdue1 votes

{quoted}

How are you getting a 44% improvement for first inhib?

your expected LP/Min for surrendering at 20 is -1 LP/Min. Your expected LP/Min for playing it to 30 minutes, even with only 8% chance to win, is -.56 LP/Min, a difference of .44 LP/Min, or 44%.

eaglei31 votes

{quoted}

your expected LP/Min for surrendering at 20 is -1 LP/Min. Your expected LP/Min for playing it to 30 minutes, even with only 8% chance to win, is -.56 LP/Min, a difference of .44 LP/Min, or 44%.

Ok, that is what I thought you did, in which I disagree. When you set the game average to 30 minutes, you are setting the time for 1 game to be exactly 30 minutes. This means to get your accurate comparison, you need to change the surrender time to reflect a 30 minute avg game as well, even if the surrender takes place at 20 minutes. This gives your lp/min of a surrender as -.67 lp/min per game. This produces a difference of about 16.4%, not 44%.

Subdue1 votes

{quoted}

Ok, that is what I thought you did, in which I disagree. When you set the game average to 30 minutes, you are setting the time for 1 game to be exactly 30 minutes. This means to get your accurate comparison, you need to change the surrender time to reflect a 30 minute avg game as well, even if the surrender takes place at 20 minutes. This gives your lp/min of a surrender as -.67 lp/min per game. This produces a difference of about 17.8%, not 44%.

Why would you consider time not played in the game as part of your calculation for LP/Min? If anything it would be closer to what was identified in a previous post, where in you would consider the 10 minutes of the next game that you were able to play sooner, and that would be based on your average LP/Min based on average game length and your average gain per win / loss per loss.

eaglei31 votes

{quoted}

Why would you consider time not played in the game as part of your calculation for LP/Min? If anything it would be closer to what was identified in a previous post, where in you would consider the 10 minutes of the next game that you were able to play sooner, and that would be based on your average LP/Min based on average game length and your average gain per win / loss per loss.

Because your entire scenario assumes the game time avg is 30 minutes. That makes time a constant. It can't be a variable for calculation purposes. Just because the time is not spent in game does not mean it does not pass.

I mean I can come back and say that a 25 minute game with an inhib loss has a -.672 lp/min. I am now changing the constant and throwing all calculations off.

Subdue1 votes

{quoted}

Because your entire scenario assumes the game time avg is 30 minutes. That makes time a constant. It can't be a variable for calculation purposes. Just because the time is not spent in game does not mean it does not pass.

I mean I can come back and say that a 25 minute game with an inhib loss has a -.672 lp/min. I am now changing the constant and throwing all calculations off.

I chose 30 because that's the average game length as per the data available. You can run the analysis with whatever you expect the game length to end up and you'll come up with similar numbers, especially if you then account for the value of that time relative to the next game.

Jesus is Savior1 votes

The numbers are actually even higher, because the stats don't take into account that a lot of those "losses"in the stats were "winnable surrenders" themselves.

Subdue1 votes

[{quoted}](name=I Follow Christ,realm=NA,application-id=3ErqAdtq,discussion-id=L1i6IjMu,comment-id=0006,timestamp=2018-03-13T23:10:58.789+0000)

The numbers are actually even higher, because the stats don't take into account that a lot of those "losses"in the stats were "winnable surrenders" themselves.

Yea, that's very true. A lot of people prematurely surrender games that could have been won.

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