Gameflow: Some pseudomathematics about comebacks, marginal advantages, and increasing entropy in competitive games

Original article here.

This article wants to explain some important factors when designing a gameflow, especially for strategy games or MOBA's. Some good points were already made here, while this article tries to extend the topic and give some intuitive-mathematical background.

For your usual strategy game (RTS or Moba to be 'interesting', there are several factors that absolutely have to be there. Like a hygiene factor (as in motivational psychology) or a necessary condition (as in mathematical proofs), without it, it's definitely a bad game, but the existence of the factor doesn't make the game good. I especially want to explain gameflow and the impact solid strategic moves should have on the game state. For this, let's imagine a very, very rough abstract game state: A player's estimated chance to win at a certain point in the game. In a two-player game, one player's win chance equals one minus the other player's win chance. If we separate a game's timeline roughly into early game, mid game, late game and finish (with the thumb rule 'early game gives advantage, mid game strengthens or turns advantage, late game decides match'), a match of a strategy game might look like this:

Graph 1: Example of a gameflow

After a head-start for blue (ok ok it's violet, but it fits so nicely in the color scheme and I will not call it violet. Period.), after the early game the red player stabilizes and overtakes blue, to crush him in the late game and finish with a win.

This is just one example; after the mid game, blue might also build his advantage and drive the win home. The question I try to answer here is, for a game to be interesting, how much should a player's advantage grow or fall, depending on the quality of his moves? There are very few correct answers to this question. A strategy game has to be designed in a way that its gameflow has the “correct” feel, as described below; otherwise it becomes uninteresting quickly.

1. Every part of the game should have a significant impact on the outcome.

 

Ok, let's develop a very simple game without specified mechanics and several distinctive game states. You start at the start, simple enough, and it's balanced, so both players start at the 50% win chance. Now for this example let's assume, that from the start of the game to what one might call the early game, one of the players gains a distinct advantage. In the graph below, the blue players win chance grew by 20 percentage points, while the red players win chance dropped by 20 percentage points. Great! The blue player has a head start, but the game is not decided yet.

Graph 2: Good example for an interesting early game

Now for a bad example. Imagine if you played StarCraft, but in the first 10 minutes you were not able to scout or attack or interact with the other player in any way. These 10 minutes would look exactly the same every time. You could write a bot to play these first 10 minutes for you and go do something else in the meantime.

Graph 3: Bad example for the gameflow of an early game. Too little impact of the first minutes.

Please don't waste my time, game. I don't want to play ten minutes for a maximum of 5% difference in win chances.

Or, the other way around. Imagine DotA with 2 leavers on the enemies side, but Techies is in their team. You virtually have won. There is practically no way to lose (if your team isn't complete non-geniuses. Which happens.). BUT the game still takes FOREVER. You have won, but you will get your win only after 30 more minutes of playing against Techies and his mines.

(By the way, after years of LoL, I swapped to DotA to only play Techies. They´re assholes and they force others to play minesweeper instead of DotA. I love them so much.)

Graph 4: Bad example for the gameflow of a late game. The game is already decided in the mid to late game.

Really, don't waste my time, game. The match is won between mid and late game. I don't want to play longer. I won. Give me my win now.

So: Every part of the game should have a significant impact on the outcome of the game. Don't waste the player's time on earth.

2. If playing “correctly”, a player should always be able to build marginal advantage.

Sort of a derivative of the first point. If you're leading, you should be able to do something with that, and build out your advantage.

Graph 5: Good example for the blue player gaining a marginal advantage over the course of a match.

In this example, the blue player is building his advantage in every part of the game; gaining relative strength with every step. Imagine that in some parts of the game the winning player could only fight to not lose advantage, but not to gain anything. The optimum for that player would be keeping the status quo. Good moves by the winning player would not be awarded with an even bigger advantage; but bad moves by the winning player would be heavily punished. This is extremely unmotivating. Some racing games use a mechanic called “rubber banding”, where especially the A.I. receives speed buffs and debuffs depending on their position, so that the field doesn't break up too much and the game stays “interesting”. This is extremely annoying though if you're driving in the first place for most of the race and make a mistake near the finish line. If the AI didn't receive their 'unfair' boost, you would have made enough time so that this mistake wouldn't cost you the game. Thanks to rubber banding though, the AI stayed on your bumper for the entire game and the simple mistake, that wouldn't have mattered anywhere else on the track, costs you the win.

I hate racing games.

And, comparing graph 5 to the graph 4 above, there is one important difference:

3. At any point of the game, there should be a possibility for the player at a disadvantage to still win the game.

There has to be the possibility for a comeback. Otherwise, why would the losing player still want to play? Of course, it should be hard to make a comeback and the probability should be small, but the realistic possibility should exist. In graph 5, the losing player gets into a worse and worse position, but even in the late game, there still is an about 1 in 6 chance that he wins. That, in my opinion, is a chance worth taking. It's unlikely, but it's not unrealistic. It happens on a regular basis, if playing often enough. But most of the time, the better player still wins. Compare this to the bad example in graph 4: Why would the losing player even bother to play out the late game when he can't win.

Now, the model I built here has an obvious disadvantage; because usually the advantages are of a statistical, and not discrete nature. This means that even if the gameflow is well designed, that sometimes the game is actually won before it is over. For this case, a 'Concede' option should be available for the losing player.

Ok, now for some more mathy observations. The next two points work together.

4. The marginal advantage gained by the winning player should decrease depending on the already accumulated advantage.

Imagine a player gaining an advantage in the early game, and playing solidly to the mid game, so that he builds out his advantage, as shown in graph 6:

Graph 6: Marginal advantage after good early game

Now imagine an even better early game, and a solid play to the mid game:

Graph 7: Marginal advantage after an even better early game.

While in the second graph the winning player has a higher advantage due to the better early game, the slope of the advantage gained between early and mid is actually lower. This can be better observed in graph 8:

Graph 8: Different slope and height of marginal advantages gained depending on the already gained advantage

This difference in slope is very important and the main counter to snowballing. The term 'snowballing' derives from the graph of an avalanche that starts with a small snowball. One small advantage results in a bigger advantage that results in an unlosable situation for the snowballing player.

The related observation is:

5. The marginal advantage gained by the losing player should increase depending on the already accumulated disadvantage.

This is the main enabler for comebacks. Again, imagine a bad early game, but now the losing player can gain some advantage in the mid game due to solid play.

Graph 9: The marginal advantage gained by a losing player in the mid game after a bad early game

And the same for an even worse early game:

Graph 10: The marginal advantage gained by a losing player in the mid game after an even worse early game

While in graph 10 the losing player still is at some sort of a disadvantage, in graph 9 he actually gained the upper hand. But, when comparing the slope of both lines, the marginal advantage gained in graph 10 is bigger than the marginal advantage gained in graph 9:

Graph 11: Comparison of the marginal advantages from two losing positions

Imagine this were the other way around, and the marginal advantage gained from the worse position were lower. Then the gameflow would look like this:

Graph 12: Bad example of a too low marginal advantage gained from a losing position

Even when playing solid in the early to mid game, the losing player still is at a significant disadvantage. A solid comeback in this kind of setup is very hard to pull off, as it takes very long to really gain the upper hand.

While we're comparing slopes, there is one very important concept to keep a game interesting. I really like the name for that. It's called:

6. Increasing Entropy: The marginal advantages gained later in the game have to be bigger than in the early game

This one sort of works against number 4, so we'll use a neutral example. In graph 13, we're starting at a neutral position early in the game, and the blue player gains an advantage.

Graph 13: The advantage gained early in the game

This is a solid advantage, and opens up to an interesting mid game. It is easy to imagine, that the development of the gameflow from this point on can have some dramatic moments.

Ok, let's look at the same advantage gained later in the game:

Graph 14: Bad example of the same advantage gained late as early

Imagine the game turned out to be a sort of tug-of war, and in the mid game neither party has an advantage. Then, in the late game, the blue player does a good strategic move and gains the upper hand. But the marginal advantage gained in graph 14 is too little. There is hardly any time left in the game, so the game should be decided equally in the late game as well as in the finish. But in graph 14, the late game can not decide anything; the advantage gained is just too small to give an interesting impact. A player would be wise not to use all his resources in the late game, because the last part of the game is much more important. Here, take a better example:

Now this looks much more interesting. The blue player gained a heavy advantage; winning between the mid to late game really paid off. The red player might still win, but he is in a really bad position. This looks much more interesting than graph 14.

...

That's it, I think. One surely could nitpick and find more things. Actually, if you find more, please comment.

Now, how does one translate this into a game?

1. Every part of the game should have a significant impact on the outcome.

This one is very intuitive. If playing your game several times and you find elements that you always do the same way (and every play tester does the same way), try to either cut it out (sometimes hard) out or give it more important decisions (most of the times harder). If you find a part that is more of a chore and doesn't actually impact the game, change that.

2. If playing “correctly”, a player should always be able to build marginal advantage.

and

3. At any point of the game, there should be a possibility for the player at a disadvantage to still win the game.

and

4. The marginal advantage gained by the winning player should decrease depending on the already accumulated advantage.

and

5. The marginal advantage gained by the losing player should increase depending on the already accumulated disadvantage.

These ones are quite hard to design correctly. On the one hand, you want the better player to slowly gain advantage. If it were simple though to turn the tables, the winning player really didn't have an advantage in the first place. Imagine a racing game with one sharp curve before the finish line. Sharp curves are excellent places to overtake an opponent. Imagine the guy in the first place, with the second one right on his tail. Because of the sharp curve, the second guy will very likely overtake the first guy and win the race. This means the second guy is actually already in a winning position, even if he is not leading the field. If your 'advantage' can simply be taken away, it's not really an advantage.

On the other hand, you want the losing player to fight and actually possibly win the game. So, make sure, that in the unlikely case that the losing player wins a battle (in contrast to the entire match) against the winning player, that the losing player now really is in a much better position than before. Most of the time, because of the structure of games, the mechanics that give a slight marginal advantage to a winning player are the same mechanics that have to give a huge marginal advantage to the losing player. If the structure of the strength gaining mechanic does not behave this way on its own, make sure that the actual battle, in which the advantage is gained, burns a lot of resources and strength.

6. Increasing Entropy: The marginal advantages gained later in the game have to be bigger than in the early game

Increasing entropy means that with a later game the game's elements becomes more game changing. The game has to be quicker in some way. A very good article with examples for increasing entropy is written by Sean 'Day9' Plott here. Basically, have more of what changes the game state (like damage) and less of what hinders changes in the game state (like HP or armor). Make the game more dangerous and give decisions higher rewards or higher penalties, so that the end of the game is reached quicker. Take care though, that this is in balance with the advantages gained from the early game. If your late game is too chaotic, what happens in the early game hardly matters anymore.

So, yeah! Gameflow. Make it interesting and worth the players' time. In my opinion, there is not a lot of deviation and playroom from curve of above's good examples; but of course, the curve is very abstract and can be interpreted for many games.

I hope this has been helpful (or interesting. Or amusing, in a 'dear-god-what-nonsense-is-this' kind-a-way.) for at least one person, then my day has not been a waste. Have a nice day yourself!

Rince