Hey all. We’ve got a guest article from Sylow this week. He’s a math major from the UK and has done some great work looking at decision making in League. Enjoy.

Introduction

Let me welcome you into the situation I want to consider. I’m playing a game of solo queue as Jax top against Irelia. It is eight minutes into the game and we are even on creep score and both level six. I find a lucky opening, jump in and secure a 1v1 kill, leaving me at 200 HP. The creep waves are pushing slowly towards irelia’s tower and the enemy jungler is off the map, but will surely kill me if he arrives before I leave. What is the best decision for me to make in this situation?

In this piece I’m going to attempt to model this situation with a little mathematics in order to show that the correct decision to make in this situation may in fact play out to lead to my death in any specific game. The TL;DR? League is a game of averages and every so often even the best player loses to Lady Luck.

The Decision

In reality, League is a complex game and there are many different options for the Jax player to choose from in this situation. Instead of trying to jump in at the deep end, we can begin by constructing a very simple model. This is what we will do in this section in order to illustrate the modelling process. Although no model will be perfect, keep in mind that so long as it is captures some of the complexity of our situation we will have something theoretically useful.

Here is the decision I want to consider:

• We have 1 player (Me, as Jax).

• He has 2 options – to push or to base.

• There are 2 outcomes – I either die or don’t die.

Now we have to specify how much I “win” in each of the 4 situations. In other words we have to be able to give a number to each possible outcome for me. This is mathematically referred to as utility, but to avoid scaring anyone I’ll refer to this as our winnings and you can think of this as a numerical measure of our advantage gained. To begin with I am going to just make up some example numbers (in the table below) for winnings and we’ll look at estimating these more accurately later on.

Finally, what is the chance that the enemy jungler is in a position such that he will kill me if I push? You could easily argue here that the jungler either is or isn’t top, but this reasoning does not help me in my decision making. Even the best player in this situation has to guess whether the jungler will gank them if they push. Technically speaking this is to say that League is a game of imperfect information, due to fog of war. However, if we were to see this situation occur in thousands of very similar games it seems likely that a pattern would emerge. If I push, a certain proportion of the time the jungler will come and kill me and the rest of the time he won’t be in a position to. If we agree that this is some fixed number, then we can model it as a probability P between 0 and 1 and can then determine what on average is the best decision to make (Note that this immediately gives us that the probability we don’t get ganked is simply 1-P).

With this in hand we can then do some mathematics to determine the optimal decision in our model, We do this by taking the expected value or average winnings of each of our two decisions. If our model were accurate, we could conclude that the expected value is exactly what the name implies, the amount we expect to win on average by taking each strategy. Although in the short term this may not turn out to be the case, if we were to play thousands of games taking each decision, we are practically guaranteed to average the expected value of that decision by a beautiful piece of mathematics called the law of large numbers. This is not all however! The best part about expected value is that in these sorts of cases it is very easy to calculate. All you do is sum all the winnings of the possible outcomes of a strategy, each multiplied by the chance of this occurring.

Let’s illustrate this by doing the sums for our hypothetical situation:

The expected value of the decision to push here is the probability of dying (given that we pushed) multiplied by the winnings in that case, plus the probability of not dying (given that we pushed) multiplied by the winnings of this outcome.

Mathematically we get:

E(Push) = (-5)*P + 10*(1-P)

E(Push) = 10 -15*P

The big E, as you might have guessed, means expected value. By running the same calculation on our other decision (assuming that we know for sure that we can base without dying) we get the expected value of the decision to base as:

 E(Base) = (-10)*0 + 0*1

E(Base) = 0

Finally we can say something about what the correct decision is by comparing these two expected values. We know we should make the decision which holds the highest expected value, so we only need ask, when is: 10-15*P>0 ?

Running the algebra we get:

10-15*P > 0

–> 10 > 15*P

–> P < 2/3

Therefore (if you were to agree that our model is accurate) you could conclude that if you think the chance of the jungler ganking is less than 2/3, you should push the lane. Similarly if you think the chance of the jungler ganking is greater than 2/3 you should definitely base. In the case that you think the chance of being ganked is exactly 2/3 (but what are the chances of that?), the model says you will do as well on average taking either decsion.

Winnings

One of the major criticisms of the model presented so far as an approximation of the scenario discussed in the introduction would be that our winnings were arbitrary. There are also a host of other criticisms one could make and I invite you to think not just of why they invalidate my model but of how you could incorporate things I have ignored into a larger, more accurate model. In this section I will try to do this a little myself, by looking at how we could come up with a more justifiable system of determining what was called winnings.

The most obvious way to do this would be to try to convert all resources into gold and then use gold gained (or, somewhat equivalently, gold denied to Irelia) as our measure of winnings. Some of these calculations are easy. We can look at the cost of buying a longsword (400g) and divide by the stats it gives (10AD) to suggest that each point of AD is worth 40g. Performing this calculation for each of the basic items would give us a conversion rate for stats into gold. How then do we turn experience points into gold? For a specific champion (in our case Jax) we could do this by determining his stat increases up to level 18, turning these into gold and then dividing by the amount of exp needed to reach level 18 (19720) to find the gold value of each experience point. Doing these sums we find:

Multiplying these numbers together and adding them we find that for a level 1 Jax to buy his way to level 18 should cost 10561 gold. Then each experience point, to Jax, should be worth 10561/19720 ≈ 0.53555 gold. It’s worth noting here that this figure, in my opinion, is not too unreasonable. While this statement may not seem mathematically rigorous, thinking like this is always a good reality check when modelling a situation you know well.

Now we can return to our model. Let’s assume that pushing in our situation will allow me to gain both gold and experience from one wave of minions (Being extremely proficient at last hits makes maths easier as well as winning, who knew?). Further let’s assume Irelia will miss a whole wave completely. At eight minutes the minion wave hitting the lane is worth 309 experience and the minions are worth 126 gold. Under our assumption that I get that and force Irelia to miss it, converting the exp to gold shows that pushing the lane gives a gold swing in my favour of:

[(309 * 0.53555) + 126] *2 ≈583

This of couse assumes that I do not die. If I die I might expect to miss a wave myself, equaling out the gold and exp missed by Irelia. Further, I will give the jungler 300 gold and 622 exp. Although we have not specified who the enemy jungler is, assuming exp on him is worth the same as exp on Jax (just to save on the calculations), we can do the sum to show that pushing and dying is worth:

[(309*0.53555)+126] – [(622*0.53555)+300] ≈ -342

Within this analysis, basing and not dying is worth 0, as it resets the lane in the middle and so leads to no gold swing eithe rway. Finally, although it doesn’t matter as we assign 0 chance to it happening. basing and dying would just give the enemy jungle the gold and exp, worth:

-[300+(622*0.53555)] = -633

Now we can produce a second table for gold advantage, which I previously called winnings in lieu of this more accurate measure, for each outcome:

Running the same calculation we did previously with our new winnings (I’ll leave this for the keen reader) we find that we should push when P<583/925 ≈0.63 . Again we should definitely not push if we guess P is larger than this. Although this model now has some justification for it, it is clearly imperfect. We have not, for instance, taken into account the value of skill points given by levels.You could even reasonably go as far as to say that this is such an oversimplification that the number we have discovered, 0.63, isn’t of any value and I would be inclined to agree with you to some extent. I hope nevertheless that I have illustrated how the model can be improved until it is sufficiently accurate to be useful and therefore that this method can give a reasonably good framework for our thoughts about decision making in League.

Conclusions: Bad Decision or Bad Luck?

This method may seem crazy and impractical, but I am not suggesting that you need to do these sums in order to make the decision in the moment. In practise, I think most people do a calculation like this in their heads for all kinds of scenarios in League anyway. They ask what is most likely to happen and what they gain or lose in each scenario and try to balance things up. The advantage of looking at things in this probabilistic sense though is that it puts the random nature of outcomes in League at the forefront of our minds.

While braving solo queue I often see people reason in this way – Jax killed top, pushed, was ganked and died. Therefore pushing top was a bad decision. Thinking along the lines of the model presented here, this reasoning makes no sense. Certainly his decision turned out badly, but it is quite possible that the top wave he got before he died was worth a lot of gold and that, across the thousands of league games going on as you read this in which similar situations occur, the Jax’s who push are not often ganked. In this case, the reasoning presented above is plain wrong – the decision to push may in fact have been the correct decision and Jax turned out just to be unlucky.

Most people seem to dislike this thought and it’s true that it seems powerless. How am I going to criticise that feeding Jax if not by observing that he has just died? Somehow we feel that it should be encumbent on all of the players in our games to not make risky plays and avoid feeding, and that if they cannot do this they are “playing badly”. If, however, you accept that a mathematical model of the type presented in this article governs such situations, there are only two criticisms you can make of another player’s decision making. You can claim that the Jax has incorrectly evaluated the relative gold advantage of pushing compared to dying (his ‘mental’ gold advantage table is wrong) or you can say that Jax did not recognise how likely he was to be ganked (he ‘mentally’ mis-evaluated P). What I think is certain is that luck plays a part in all League games and should prevent us from making cut and dry statements about good or bad play in any particular game.

In conclusion, worth is really not something to be thrown around in all chat. Which decision is worth the most (on average) is a central question in decision making and is usually decided in any particular game based on incomplete information. Questioning what is worth seriously should allow you to make better decision and ultimately win more games, regardless of what happens in the one you are in at the moment.