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Thoughts on game complexity

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I was thinking recently about one way of making a game fun, and I think that one way is to make it so that it is possible to estimate the quality of a chosen strategy, but not with 100% accuracy. If it's always possible to reject the wrong strategies then it becomes quite boring (eg. tic-tac-toe), and if it's impossible to pick a better strategy (eg. rock-paper-scissors) then it's also not much fun. So there's a sweet spot in the middle - but how do you create it? Extra complexity seems to be one route, as taken by games like Civilization - so many choices and options that it's hard to predict what is the best one, though it's easy to see that some choices are better than others most of the time. Is there another way that doesn't just involve throwing so many choices at the player that mentally managing the decision tree becomes unwieldy?

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Randomness is always useful in adding that element of unpredicibility into the mix.

There's also the chess approach; have relatively simple options but involve planning to great depth.

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Randomness ultimately averages out though. It may double the decision tree if there are discrete successes or failures but it doesn't really add significantly more complexity into the equation.

As for chess, that's sort of what I was talking about, though I don't really agree that the options are simple given that you tend to have 20 to 30 different moves you can choose at any one point. Many computer games offer you fewer meaningful choices than that (eg. in which skills to choose, which items to wear, which buildings to construct).

And where does the requirement to plan to great depth come from? If that could be expressed clearly, perhaps it could hint at ways to add strategy to some games that is better than merely finding a new permutation of stat bonuses.

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Original post by Kylotan
As for chess, that's sort of what I was talking about, though I don't really agree that the options are simple given that you tend to have 20 to 30 different moves you can choose at any one point. Many computer games offer you fewer meaningful choices than that (eg. in which skills to choose, which items to wear, which buildings to construct).

Well chess is very predictable, which is why you can think so many moves ahead. With a game things tend to be unpredictable (whats round the corner? Will I need to save this item for later?). I guess that this means that the complexity can be somewhat lower but still interesting.

Of course if you go too far with this then it ends up being impossible to make a good decision and becomes unrewarding. Eg. most RPGs which force the player to make long ranging decisions about the character with little or no knowledge of what will be useful or not (Eg. trying to pick between swimming or sneaking skills with no idea whether you might spend 90% of the game sneaking).

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Forgive me if I was missing the point; I've got a lot on my mind and this topic is extremely abstract [smile]. To be honest, I'm starting to get confused if we're talking about A.I. or just game theory; I'm assuming it's the latter.

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Original post by Kylotan
Randomness ultimately averages out though. It may double the decision tree if there are discrete successes or failures but it doesn't really add significantly more complexity into the equation.

Well, it depends a bit on exactly what the randomness is for, and the result of a success or fail. I tend to like randomness in games because it's a simple way to add a lot of implicit possibility; you need to factor in a lot of different options for every random element. It's also great because psychologically people aren't very good at dealing with averages with random elements; it's the reason the lottery is so successful.

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As for chess, that's sort of what I was talking about, though I don't really agree that the options are simple given that you tend to have 20 to 30 different moves you can choose at any one point. Many computer games offer you fewer meaningful choices than that (eg. in which skills to choose, which items to wear, which buildings to construct).

Yes; I was thinking of picking something like draughts (checkers) as my example, but chess is more canonical. Usually though a large proportion of those 20 moves can be dismissed for being irrelevant or stupid. I can somewhat agree that the number of options given to you is greater in chess than many games, but the comparison is hard to make in the general sense, particularly if we're comparing turn-based board games versus real time.

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And where does the requirement to plan to great depth come from? If that could be expressed clearly, perhaps it could hint at ways to add strategy to some games that is better than merely finding a new permutation of stat bonuses.

I wouldn't say the ability to plan to great depth is a requirement; I don't play chess that way because I'm not very good at the game. But I was thinking that to be really good at chess you need to go beyond the next couple of moves and understand the interaction between the pieces at a much deeper level.

I admit I'm not an expert at how the experts play chess, so I'm not sure if "planning at great depth" is the best way to describe how they function. I do have a mate who plays at international level; I should ask him because now I'm curious.

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I'd say, play more with risk versus reward. We all love low-risk, high-reward choices: cheap splash damage towers in those TD games, Overlord tanks with gattling guns in Generals, rapid fire snipers in CSS, and so on.

Rock-paper-scissors is pretty strong in that regard: everything is high-risk, high-reward. For experienced players, there's a lot of strategy involved: the way the opponent acts, reacts, how he played in previous games, and so on - all these factors play a role for them. There's a psychological game going on between them. That may just not be as visible to us, which is probably why we don't like it as much.

I also think dependencies between choices play an important role. In chess, units can cover each other, in TD games, slow towers can keep enemies within range of high-damage towers for a longer time (or distribute them more evenly across your towers, for better efficiency, etc.), in RTS games, different units can cover each others weaknesses, etc.


In other words, I think it's not so much the number of decisions that counts, but the depth of their effect. Different risks and dependencies will make it harder to foresee the consequences of your choices.

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I would recommend reading "Game Architecture and Design" (by Andrew Rollings and Dave Morris). Especially the first part as it deals with this issue.

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Extra complexity seems to be one route, as taken by games like Civilization - so many choices and options that it's hard to predict what is the best one, though it's easy to see that some choices are better than others most of the time.

There is Complexity and Complexity. One for of complexity is to dump a lot of choices on a player (like in civilization). However, another form of complexity is to have a lot if interactivity between the choices (like in Chess).

I am in favour of simple elegance, so I come down on the side of Chess. I like to have a few choices, but have those choices have a high level of interactivity with each other. The reason I like this form of complexity is that it more reliably leads to emergent gameplay, than just having a lot of strategically shallow choices. However, some people like having lots of choices, but I am not.

It is Depth vs Breadth.

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if it's impossible to pick a better strategy (eg. rock-paper-scissors) then it's also not much fun.

I agree. As it stands R/P/S is not a fun mechanic by its self. However, you can start to make it more interesting and fun by slightly breaking the symmetry of it (This is what "Game Architecture and Design" talks about).

So to start off with the RPS game has 3 choices and if you guess right, you win. Lets call winning a value of 1 and the cost of playing one of the options (Rock, Paper or Scissors) 2/3. The reason I call it 2/3 is that you have to choose 1 of 3 so the cost of playing 1 is not to play the other 2.

This is symmetrical. None of the choices have a higher cost or better score than the others. Now lets break the symmetry.

Lets say the costs for playing are Rock: 3/4, Scissors: 2/3 and Paper: 1/2 with victory remaining as being 1.

Now the whole game is different. Paper is the cheapest to play, but Scissors is only slightly more expensive and Rock being the most costly.

Because there is now different costs involved, it allows us to get a handle on what the other player might be thinking: "Kylotan doesn't have many points left, so he might try to play it safe and chose paper, but he still has enough points to play Rock. If he goes for the cheapest choice, then I can win by playing Scissors. I know that Kylotan is a sly player and would know that, so I think that he will choose to play Rock even though it is the most expensive because he thinks that I will choose scissors to counter the safe choice of scissors..." and so on.

If a player has a lot of points, then their choice can be more random, but if points are low, then their choice becomes dictated by the environment (the number of points) and the player's psychology (like in poker). It allows for bluffing and second guessing you opponent more than the perfectly symmetrical R/P/S mechanic.

If this was a Real Time strategy game, you might have Archers cost 6 Gold, Pikemen cost 8 Gold and Knights cost 9 gold. If 1 Knight can beat 12 Archers (before dying), 1 Pikeman can beat 12 Knights and 1 Archer can beat 12 Pikemen (then you have exactly the same system as I presented above - just multiply everything by 12, both costs and wins).

You can then further break the symmetry by adding in a few other units that don't directly effect the outcome of the contests (say a healer or a peasant that mines for gold) and further strategies and tactics will emerge (should you go after the peasants, or the healers, or will you have to have a pitched battle between the main 3 units, moving then to outflank the enemy).

Lastly, you can add in choices that mess with the dominance of the choices (so sometimes scissors might just be able to beat Rock and paper can manage an uprising against those oppressive scissors). In the case of the medieval units, sometimes the archer might not be effective against pikemen (by placing the pikemen on top of a hill, etc).

So, 3 techniqies that can help:

1) Break the symmetry slightly (with costs and rewards) between your choices.

2) Add in options that don't directly (but indirectly) effect the outcomes of the contests between your choices.

3) Add in options that allow you to change the dominance of the choices.

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Let me try and elucidate a bit - this is necessarily an abstract topic, as I'm looking at the fundamentals that underpin all games to a lesser or greater extent. But I'll be as clear as possible.

Firstly, I'm trying to ignore randomness because it can be 'rolled into' the mathematics. Compare one chess piece taking another, with a battle in Civilization - say, a militia attacking a phalanx, who will win just one time in three. The random factor only adds 1 extra outcome. In chess, you can choose to take the piece or not. In Civilization, you can choose to attack and success, choose to attack and fail, or choose not to attack. Now, the relative importance of those decisions does rest on the random factor, but the same would apply in Chess since you don't know what the opponent will do. So the difference it makes to the decision tree and hence the planning is not all that great.

Trapper Zoid - when you say "Usually though a large proportion of those 20 moves can be dismissed for being irrelevant or stupid." you're totally right, but a new player might not be able to dismiss more than one or two, and even for an expert player, the top 5 or 6 are hard to rank accurately. And that's exactly the middle ground I think is 'right'... a decent player has an idea of what is good and what is bad, at least in a given context, but is unable to know the theoretical perfect strategy that would make playing trivial.

I think great chess players see overall patterns that substitute for needing to look at each potential move exhaustively. It's a heuristic they've developed to judge move quality without needing to explicitly plan. Is there a way to design a game such that players need to develop that sort of holistic understanding, and if so, does it require that you set up such a large branching factor to the decision tree?

Captain P - the dependencies between choices is a large part of the puzzle, I think. But how can that be explicitly engineered in a game, beyond something trivial like "wear the complete matching suit of armour and get an extra +10"? It's easy to recognise in existing games but I'm not sure how to implement it without it being a crude tool. Depth is easy to recognise, but how is it created?

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My previous post was done before I saw this, so I'll answer separately.

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Original post by Edtharan
I would recommend reading "Game Architecture and Design" (by Andrew Rollings and Dave Morris). Especially the first part as it deals with this issue.


Aargh, I've got that book in my 'to read' pile. I may just promote it to the top of said pile now. I was hoping I'd find some books or papers on this sort of topic, so thanks.

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I like to have a few choices, but have those choices have a high level of interactivity with each other. The reason I like this form of complexity is that it more reliably leads to emergent gameplay, than just having a lot of strategically shallow choices.


The problem with 'emergent' properties is that it's hard to create something that manifests them in the first place. :) How would you go about creating something that gives as much depth as chess, without merely taking chess and altering a few rules?

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if it's impossible to pick a better strategy (eg. rock-paper-scissors) then it's also not much fun.

I agree. As it stands R/P/S is not a fun mechanic by its self. However, you can start to make it more interesting and fun by slightly breaking the symmetry of it (This is what "Game Architecture and Design" talks about).[/quote]

I will read more on that for myself. I agree that asymmetry can make things interesting, although I'm having a bit of trouble imagining how it would apply outside of scenarios where you're pitting your resources directly against someone else's. An example I keep thinking of is choice of equipment or spells/skills in an RPG-like game - how do you make such a choice interesting, and make it so that - given the same resources available to all - good players will be able to pick better combinations, but not necessarily be able to know which is mathematically the best?

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theres the "fog of war approach"

in that you have a fairly predictable system but you hide some of the information from the player and force them to build a strategy on what they know while considering everything they don't know

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An example I keep thinking of is choice of equipment or spells/skills in an RPG-like game - how do you make such a choice interesting, and make it so that - given the same resources available to all - good players will be able to pick better combinations, but not necessarily be able to know which is mathematically the best?

You are right, the S/P/R system is mainly good for direct confrontations. In a game of spells and weapons, this S/P/R system will mainly apply during the confrontation. However, indirect confrontations and also befit from S/P/R, but it is harder to apply and you have to approach it more abstractly.

A batter system for non confrontational choices might be through a Cost/Benefit system, where each choice has something that is good, and something that is bad. This way the player has to deal with the bad to ge the good, and if they can only choose one, then they have an interesting decision to make (that is what reward do they want and at what cost?).

Also (a bit off topic), here are a couple of wikipedia articles that might be of interest:

http://en.wikipedia.org/wiki/Game_theory
Game theory is always a good place to start.

http://en.wikipedia.org/wiki/Prisoner's_dilemma
A good place to start when trying to understand game theory

http://en.wikipedia.org/wiki/Diner%27s_dilemma
An interesting modification to the basic prisoners dilemma

http://en.wikipedia.org/wiki/Ultimatum_game
An interesting game that when performed in real life (using repetitions among people that communicate with each other) shows the origin of Altruism (that the cheats get punished).

http://en.wikipedia.org/wiki/Tragedy_of_the_commons
It would be interesting if this could be applied to an RTS game.

I think that the Ultimatum game and the Tragedy of the Commons might create a new and interesting way to give resources to players in an RTS game (imagine an RTS where the resources are split between you and your opponent through an ultimatum game mechanic, or one where you share a common resource that regenerates slowly, but if it is depleted it can't regenerate).

Or you could weave a prisoners dilemma into an RPG plot.

The advantage of understanding Game theory, although it is not always useful for actual games (it is better to economics and politics than games for fun), is that it gives you a tool to craft and test situations where players have to make choices, and it can lead to interesting applications of them in a game with a bit of imagination.

By using game theory to help craft a set of choices available to the player you can control the expected pay-offs and penalties associated with a set of choices and easily see and balance them.

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Yeah, I'm quite familiar with economic game theory, and it's obvious that all this is underpinned by that, but it's hard to see how it applies usefully to designing games - after all, spotting the optimal strategy for those dilemmas is actually trivial once you see the 'trick', and there are typically only 2 choices, which is not good enough for most games. The complexity is too low to be fun. Perhaps the iterative nature of most games makes it more interesting, in the same way that iterative prisoner's dilemma is more interesting, but even that has a trivial best strategy too, apparently. To hide that from the player may well require more possible choices, making the payoff matrix far bigger.

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My theory for creating challenging game-play dynamics is to give the user many choices of actions (some basic, some complex... but all of which have clear results), and then force the user to optimize not only their choices, but their sequence of choices through both experience as well as trial and error. Time constraints and lead times also help improve the experience.

Example 1 - Strength vs. Economy
A user is given 2 basic choices: Build up your Army, or Build up your city. This is a classic scenario of games like StarCraft, Civilization, etc. While it is of greater economical advantage to focus solely on ones city, it is unfeasible due to the threat of war. This creates a difficult, but understandable choice a user must make in order to optimize game play.

Example 2 - Ranged vs. Melee
A user is given a choice between 2 soldiers: One is weak, but ranged. The other is strong, but fights hand to hand. Again, this is not a complicated decision per say, but the results of the choice may make the difference between victory and defeat. Unfortunately, the effectiveness of this choice is a bit random because you can never accurately predict what your opponent will do. However, I find randomness of this nature better than computer generated chance.

Example 3 - Upgrades vs. Soldiers
A user can upgrade his weapons at a large cost. This creates a moment of weakness for the player upgrading, but can result in a large advantage. What do you do? Again, there is a good bit of chance involved with this decision, but it is not computer generated chance.

From here, we keep adding simple choices that we will present to the player. There are ultimately hundreds of 'vs' decisions (death chance vs. big payoff, high vs. low, numbers vs. size, etc). The key, which is not necessarily quantifiable, is to find a way to mix all of these decisions together without overwhelming the end user. How a game designer does that is a matter of artistic style. Note: Some styles are better than others.

Also: Challenge, as I have come to learn, does not mean difficult to figure out.... rather it means easy to figure out, extremely difficult to master.

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Original post by Cygnus_X
Example 1 - Strength vs. Economy
A user is given 2 basic choices: Build up your Army, or Build up your city. This is a classic scenario of games like StarCraft, Civilization, etc. While it is of greater economical advantage to focus solely on ones city, it is unfeasible due to the threat of war. This creates a difficult, but understandable choice a user must make in order to optimize game play.


Imagine your game consisted solely of 2 statistics, City and Army, and every 10 seconds you could invest 1 point into just one of them. Would that make for a fun game? I don't think it would. You'd eventually work out the optimal balance and choose that all the time, until you got bored enough to try the other strategy.

Therefore, at the level I am talking about, the issue isn't really so much about whether you choose between building up your army or your city, but in the plethora of sub-options that let you build up both the city and the army in different ways, and to strike different compromises. Without those extra choices adding complexity, the choice seems to be uninteresting.

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Example 3 - Upgrades vs. Soldiers
A user can upgrade his weapons at a large cost. This creates a moment of weakness for the player upgrading, but can result in a large advantage. What do you do? Again, there is a good bit of chance involved with this decision, but it is not computer generated chance.


Ultimately this comes down to information - if you can tell that you won't be attacked during the upgrade, you should always upgrade, and if you can tell that you will be attacked, you should never upgrade. If you don't know, you have to multiply the factors together and see which is best. So the gameplay here really comes from the management of information - can you get information about the other player or not, and if so, how much? There are potentially interesting choices there, of course, since gathering the information typically costs resources that could be deployed elsewhere.

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Original post by Kylotan
I was thinking recently about one way of making a game fun, and I think that one way is to make it so that it is possible to estimate the quality of a chosen strategy, but not with 100% accuracy. If it's always possible to reject the wrong strategies then it becomes quite boring (eg. tic-tac-toe), and if it's impossible to pick a better strategy (eg. rock-paper-scissors) then it's also not much fun.

So there's a sweet spot in the middle - but how do you create it? Extra complexity seems to be one route, as taken by games like Civilization - so many choices and options that it's hard to predict what is the best one, though it's easy to see that some choices are better than others most of the time. Is there another way that doesn't just involve throwing so many choices at the player that mentally managing the decision tree becomes unwieldy?

Yes throwing on him even more choices, and allow him to live with these choices, and have large possibility to change them without losing.

Actually there are not any problems with complexity, there are more problems with user interface, and problems with attention deficit of the player. If he could play just simple games, yet he becomes bored quickly... There is not help for him. People like that are often result of overdose by TV networks.

There is no sweet spot in the middle, the correct complexity results from game design. Even crazy games like Harvest moon, Lufia, Harpoon, Pokémon, or SEIV have theirs players.

Of course SEIV isn't completely reasonable example, because its complexity doesn't work well together and AI can't use it correctly, in addition there is complexity in one area, however unnecessary abstraction in other area.

On the other hand Pokémon actually hid its insane complexity under story. These Pokémons need to be trained, however what level of training would they get? It depends on theirs trainer, and what they experienced. If they had easy lives by avoiding challenges, they will have harsh time but they'd make it. If they had harsh live, however theirs trainer took care about them, they would curb stomp the opposition singlehandedly. The point is not every player even notice the complexity, it's somehow hidden under scenes, these players that do notice are the ones who are more likely to enjoy a complex game. (The others could stay ignorant as they are.)

From my experience it's more about what level of complexity the developer could stomach and if he would like to play that game with such level of complexity. If it's too simple for him... Why bother to make it?

[Edited by - Raghar on July 16, 2007 1:21:14 PM]

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Original post by Kylotan
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Original post by Cygnus_X

Example 3 - Upgrades vs. Soldiers
A user can upgrade his weapons at a large cost. This creates a moment of weakness for the player upgrading, but can result in a large advantage. What do you do? Again, there is a good bit of chance involved with this decision, but it is not computer generated chance.


Ultimately this comes down to information - if you can tell that you won't be attacked during the upgrade, you should always upgrade, and if you can tell that you will be attacked, you should never upgrade. If you don't know, you have to multiply the factors together and see which is best. So the gameplay here really comes from the management of information - can you get information about the other player or not, and if so, how much? There are potentially interesting choices there, of course, since gathering the information typically costs resources that could be deployed elsewhere.


First, I believe your response to the decision of weather or not to upgrade emphasizes my point that even small decision have vital and complicated roles in the overall game experience.

Second, I'd warn a developer about emphasizing making a game complex. I have found that players can get easily frustrated over game concepts that are difficult to understand or adjust to. Again, the idea should be to develop a game with a simple pattern, but a difficult solution (Sudoku and Chess are both good examples of this).

Finally, if we're not talking simple warrior or building selection, here are a few other ideas (just ideas, not well thought through concepts) that could be employed:

Stock Market for Resources - Ie, you can trade require resources on an open market, and the trade value of each resource changes based upon demand. I haven't seen this done well in many games.

Team Contribution - You can set up a team structure such that it is 'best' for the team if each member contributes equally to a common goal, but it is 'better' for an individual to have all his team-mates contribute, and he keep his resources to himself. This would be a spin-off of the herder/pasture delimma.

Unity - This concept is employed by puzzle pirates. For those who haven't played, there are multiple stations on the ship that change difficulty based on the performance of your team-mates. If the crewman at the water ejector is doing poorly, it makes it tough on the captain (ie, more weight, less turns). If the captain is doing poorly (and is getting shot), it makes things difficult on the carpenter. If the carpenter is doing poorly, it makes things difficult on the water ejector. Thus, a player is only as good as his team-mates.

Conclusion: Good game complexity comes from unique permutations of these different concepts. Not all of them have to be included.

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There are definitely lots of ideas like that which can be employed to make a game more interesting. But is it possible to describe them more abstractly, in terms of the decision tree, or in cost/payoff matrices as in game theory? And don't these things have to be more complex or they become too trivial? eg. Both the team contribute and unity ideas are easy to find the optimal strategy for.

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You can, potentially, write down a game in a purely mathematical form, but as a completely abstract object it won't tell you much without further specifications. For instance, there is a massive fundmanetal difference between sequential (fully turn based) and simultaneous (both take turns at the same time) games, and different problems arise in either, requiring very different notation to solve. In a sequential game you have to watch out for "perfect strategies" (like in tic-tac-toe) whereas in simultaneous games these can be entirely avoided provided the game is symmetric (both players have the same choices, and the same rewards). As for whether there is a single definitive property that can detirmine how "good" a game is, I'm not too certain. The only thing I can think of is that a sufficiently superior AI (one with greater memory and processor time) can potentially beat an inferior one on the majority of occasions. This would be an intensely complicated property in an abstract mathematical setting though.

Probabilities can leave a game unchanged or drasticaly alter them depending on how they are implimented. If you had R/P/S where R beats S with 75% chance, P with 25% chance and R with 50% chance etc. then the core theory of the game is identical (in other words AIs in either game would be exactly matched*). On the other hand, in a game like Poker, the actual game is about modifying your returns for a particular round based on your estimated probability of winning. You could do a version of R/P/S using probabilities in a similar way to Poker as follows: Each player is assigned one of R, P or S randomly at the start of a round, then the players take it in turns to modify how much they want R, P and S to be worth for that round.

* - now that I think about it, this could be a good starting point. Given a game, take the set of all AIs as an abstract object S, and define a relation on S such that xRy if and only if x will beat y on average (the relation need not be transitive remember!). Then we can define an "isomophism" of games if the possible AIs from one can be mapped bijectively to the other in such a way that R is preserved.

Edit: Continuing on this idea, you could define a game as "fair" if for each x in S there exists a y in S such that yRx. In other words, for every strategy there is a strategy that beats it.

[Edited by - H4L on July 16, 2007 6:52:29 PM]

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Edit: Continuing on this idea, you could define a game as "fair" if for each x in S there exists a y in S such that yRx. In other words, for every strategy there is a strategy that beats it.

Which is what the Rock Paper Scissors game (or an intransitive relationship) encompasses (although there are other game systems that also exhibit this formulation that aren't SPR derivatives).

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Second, I'd warn a developer about emphasizing making a game complex. I have found that players can get easily frustrated over game concepts that are difficult to understand or adjust to.

I would say there is a difference between Complication and Complexity. Complication means that the game is more obscure, or there are more choices than a player can effectively consider. Complexity is that the components of the cam interact in many ways.

Chess is an example of a Complex game as it's components (the pieces, the board and the turn based structure) have a very high level of interactivity (any piece is capable of capturing any other piece if used correctly - that is placed on the board). Monopoly is a more Complicated game as there are many more components and there is less interactivity between them (on average).

Complexity and Complication, for their own sake, should not be advised (and should be advised against), but both can be used to improve a game.

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Quote:
Original post by Cygnus_X
Example 1 - Strength vs. Economy
A user is given 2 basic choices: Build up your Army, or Build up your city. This is a classic scenario of games like StarCraft, Civilization, etc. While it is of greater economical advantage to focus solely on ones city, it is unfeasible due to the threat of war. This creates a difficult, but understandable choice a user must make in order to optimize game play.

Imagine your game consisted solely of 2 statistics, City and Army, and every 10 seconds you could invest 1 point into just one of them. Would that make for a fun game? I don't think it would. You'd eventually work out the optimal balance and choose that all the time, until you got bored enough to try the other strategy.

Actually, the reason this dose not make an interesting game is that there is no real interactivity between the components. If we add in one: That the higher the level of your city the faster you produce Armies. Then it does create an interesting choice.

Now you have an advantage to upgrading your city instead of increasing your army. And because of that interactivity different strategies can present them selves.

First there is the Rush: Only build armies. In the initial scenario this was the only viable strategy as upgrading the city would not effect your army sizes (except to reduce the number of armies you have).

Second is the Counter Rush: Build enough armies to hold off the enemy armies (althoug not win), but then upgrade the city to the point where you can pump out armies fast enough to overwhelm the enemy when the do finally attack. As the Rushers won't have the ability to pump out armies as quickly, you can hold them off and then swarm then under. The difficulty here is in gauging how many armies you will need to hold off the rush long enough to give you time to build up your swarm.

The last main strategy is the Economic win: In this, if they don't rush (if they are doing the Counter Rush), and you spend all your efforts on just upgrading your cities, eventually, you will have the ability to out produce your enemy and then do a late rush and swarm them under.

Actually, unintentionally, this creates a Scissors Paper Rock relationship between the Different strategies. Rush beats Economic, Economic beats Counter Rush and Counter Rush beats Rush.

The main difference between this type of game and a pure SPR is that SPR is an all or nothing game. You have to choose one and can't easily change it later. In the cities and armies game, there exists a continuum between each of the different strategies and you can "hedge your bets". You could go for the Rush, but also upgrade your cities a little which give you a better chance to beat a Counter Rusher in that they will have misjudged the number of Armies needed to finally swarm you under.

However, you might think this would lead to the dominant strategy being to find the middle ground of all three strategies and following that. However, any one of the three extreme strategies should be able to beat this Middle Ground Strategy. But, it also pays to be flexible, so following one of the extremes make it a gamble.

How is that for complexity? Just by creating an interaction between the city upgrades and the armies, you create a complex game that has no optimal solution.

On a final note: http://www.gamasutra.com/features/20070123/chelaru_01.shtml
Have a look at this gamasutra article, especially the bit about signals and fake signals.). These can also be used to make a game have more complexity.

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Original post by Edtharan
Actually, the reason this dose not make an interesting game is that there is no real interactivity between the components. If we add in one: That the higher the level of your city the faster you produce Armies. Then it does create an interesting choice.

Now you have an advantage to upgrading your city instead of increasing your army. And because of that interactivity different strategies can present them selves.


Well, I was assuming that there was a benefit from having a powerful city too. The point I was making is that if you just have 2 variables, even if both feed into each other and both can be used in resolving whether the game is won or lost, there is an optimal strategy and it's likely to be trivial to find. If you don't know what your opponent will do, then there is likely to be one and only one choice that gives you the highest chance of victory. If you do know what your opponent will do, then you have probably been told exactly which will give you the highest chance of victory. Even adding asymmetry to the mix, while making it more interesting, ultimately can show you which choice is most likely to win, given a rational opponent and not a random one.

A lot of these methodologies rely on an opponent who is slightly random - either a human who will sometimes choose a suboptimal strategy to try and trick you (or because they're bored), or a computer that will randomly choose one, or an opponent of either type who will try to predict what you will do based on previous actions. And I think that's sidestepping the essence of game complexity that I was hoping to single out in this thread. With games like Chess, or Civilization, the optimal strategy is hidden by the level of complexity, which is a large part of what makes it fun. A lot of what people are proposing here are instead games where the optimal strategy is apparent, but where how to beat it is also apparent, leading to a bluffing war. This is certainly a valid way to design a game but it's not what I'm interested in at this stage. I'm interested in how to design enough complexity to hide the optimal strategy, without just throwing more and more features at it until it seems to work.

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Original post by Kylotan
I'm interested in how to design enough complexity to hide the optimal strategy, without just throwing more and more features at it until it seems to work.


Ok, lets consider games that currently incorporate this idea into their design. The best example I can think off would be chess. The piece movements are individually simple. The goal is simple. But the number of permutations of moves in any one game is well over a billion.

Thus, we beg the question... is there an optimal strategy to chess? Some would say yes. For each opening move by white, there are counter movements by black that lead to a higher probability of winning. For those who aren't familiar with chess, the first few opening moves are predetermined for good players. They even have fancy names such as dutch stone wall defense, queen side indian attack, birds opening, 4 horse opening, etc. With years of player experience, and with the aid of computers, all the possible opening sequences have already been optimized. However, once you get about 12 moves in, there are simply too many choices to go through to have a optimal path defined. While after 12 moves one player may have an advantage over the other, the optimal path is suddenly lost (and while it may be possible for computers to find the optimal path, i don't foresee people ever figuring it out). From here, a player must depend on his own skill, and not the predetermined moves of other players, to win the game.

Reverse Engineering
Is there a way to reverse engineer this idea? I believe there is. Chess only has 6 unique pieces with each piece having a unique abilities. It should be easy to use this idea to become the basis for other game design. I believe StarCraft is one of few games that have used this concept effectively. And yes, the first few moves in StarCraft are semi-defined. Anyone who has ever played Protoss knows how important it is to build a pylon on the 7th peon such that you can get a gateway/forge on the 8th peon in order to prepare for a rush. But, after that, there are too many variables to have a true optimal path.

Question:
Chess has enough complexity with such simple rules that it has kept a place in history for hundreds of years. However, it is not the most popular game in today's culture. Why?

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Well, I was assuming that there was a benefit from having a powerful city too. The point I was making is that if you just have 2 variables, even if both feed into each other and both can be used in resolving whether the game is won or lost, there is an optimal strategy and it's likely to be trivial to find.

If you look closely at the example I gave, there is no perfect strategy. In the game, the none of the extreme strategies are a perfect one as each has a counter and going down the middle of the road can be beaten by any of the extreme strategies. So there is no perfect solution. Even with perfect information, you can't even settle on a perfect strategy, you will have to cycle between several different strategies as you opponent tries to counter your moves (with perfect information, the game will likely descend into a stalemate depending on the switching costs to change strategies - that is how much time and other resources it take to change to another strategy).

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With games like Chess, or Civilization, the optimal strategy is hidden by the level of complexity, which is a large part of what makes it fun.

Actually, in chess there is sort of an an "optimal" strategy that can be employed, but if your opponent knows how to counter it (just like scissors counter paper), then this strategy breaks down.

This strategy give you the ability to make a check mate in only a few moves.

1) You move the Pawn in front of your King out (e2 to e3 or e4).
2) Move out you bishop to threaten the pawn the opponent's pawn (f1 to c4)
3) Move your queen out to threaten the same pawn (d1 to h5).
4) Take the pawn with your queen (h5 to f7).

This will automatically give you a checkmate unless they have move pieces out to either block you (example: e7 to e6), threaten one of the needed positions (example: g7 to g6) or allowed their king room to escape (example: d7 to d6 or d5).

It is a risky move as it places two of your powerful pieces out unprotected on the board. This would be the RTS equivalent of a rush strategy, and unless your opponent does not know how to counter this obvious strategy, you can easily win (but most competent chess players actually know this technique and it's easy counter).

So this is the chess equivalent of a Rush strategy and it is countered by a minimal positioning of pieces, which followed by a quick swarm can potentially eliminate two of your opponents powerful pieces (the equivalent to the Counter Rush strategy in the city game example form my previous post). The problem with this, is that a clever opponent might lead you into a false belief that they are going to do a rush and if you respond to that signal, then it can place your pieces into a position that is not as defensive and you opponent can exploit that to their advantage (the equivalent of the economic strategy in the city game). And of course, if your opponent tries for the long game and doesn't counter a rush strategy, then the rusher wins.

So, using this strategy set, Chess it's self presents the same (generally speaking) strategies as the City game. However, in chess, the switching costs of changing strategies is not too expensive (at worst you might loose a few pieces and take a few turns to get you pieces into the required positions). Also, with chess, these "Gambits" don't have to use all your pieces, so that even if a particular strategy fails, you might not have any costs to switch strategies.

As you learn more about chess and play it more, you learn about "Chunking", that is instead of just looking at each individual piece and its position relative to all the other pieces, you start to see larger patterns involving multiple pieces and how they influence the board (multiple pieces essentially start being parts of a single entity). It is usually at this point a chess player start to get good.

In the game concepts presented in this thread, the scenarios have been very simplified and because they are so similar to known game, the chunking for them is already known by us and so we can see these larger patters and their results more easily (that and we are specifically designing the examples to make this chunking obvious). The chunking that we are talking about are the various strategies that we are discussing.

For example, in the city game example, we didn't talk about the deployment of the armies (where a well deployed army might be able to overcome a larger but less well deployed army), but only discussed the relative strengths of the opposing players as the number of armies they have. So we "Chunked" the army deployments and assumed that both players were equally matched in their ability to deploy their armies.

If we had included that into the example. Say that, even though you are initially outnumbered, but you have the ability to chose when to attack and when to withdraw, and can do so successfully, you could withdraw from army groups that could beat you and attack army squads that you can beat. Then a player who initially chose to do an mostly Economic strategy, might just be able to beat a rusher player, even though the main rules says that this is not a good counter to a rush.

Also, to make the City game more complex, you could add in some complications, namely that the armies have a S/P/R relationship applied to them (for a medieval game: Knights -> Archers -> Pikemen -> Knights). This would then make the strategies of Rush, Defence and Economic much more uncertain and based on a player's skills and ability to adapt.

If you then put in signalling (spies) and the ability to fake a signal (counter intelligence), then the game becomes very complex and the chance of a simple "Best" strategy is almost non existent.

In fact, the Signal/Fake are used a lot in Beat-em-up games, where the gameplay is usually tightly based on the Scissors/Paper/Rock mechanic (or at least move/counter move). This faking ads a lot more complexity to the gameplay as it move the focus from the character's abilities to the person's abilities and psychology (just as chess does and it also makes us of the ability to fake signals - but not as simply implemented as in the computer game). In chess you can fake a signal by threatening pieces you don't intend to capture, forking then (moving one of your pieces so that it threatens multiple enemy pieces), and so on.

It is usually the incomplete informational aspect of computer games that means that signalling and faking are not as complex as chess. Because chess is a complete information game (you can see the positions of all the pieces all the time), and signalling and faking are by necessity much more subtle and indirect than might be in an incomplete information game (but even an incomplete information game can use subtle and indirect signals/fakes as in a complete information game).

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Well, I was assuming that there was a benefit from having a powerful city too. The point I was making is that if you just have 2 variables, even if both feed into each other and both can be used in resolving whether the game is won or lost, there is an optimal strategy and it's likely to be trivial to find.

If you look closely at the example I gave, there is no perfect strategy.


There's a difference between an optimal strategy and a perfect strategy. R/P/S has 3 optimal strategies; it just happens to be true that they are also the worst strategies. Iterative R/P/S, while no longer having an optimal strategy as such, has one optimal metastrategy - totally random choice. Any other metastrategy gives information to the opponent which increases their chance of beating you to above 1 in 3. Most iterative systems work the same way - a single optimal strategy no longer suffices but it's usually quite trivial to derive an optimal metastrategy.

Quote:
(with perfect information, the game will likely descend into a stalemate depending on the switching costs to change strategies - that is how much time and other resources it take to change to another strategy).


But that's the point - the optimal strategy probably just becomes one of the two following:
- if changing strategies is cheaper than the cost of using the wrong strategy, the optimal metastrategy is to always change so that you always have the counterstrategy ready.
- if changing strategies is more expensive than the cost of using the wrong strategy, the optimal metastrategy is to continue on with your current one, which will beat your opponent due to the cost of them changing.

There's not really much else to it - if certain strategies cost more to choose, or cost more to switch between, that factors in, but there's still ultimately an optimal metastrategy which, if employed once against an infinite number of random players, will win more often than any other metastrategy. The only hard part is gathering the data on 'costs'.

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Actually, in chess there is sort of an an "optimal" strategy that can be employed, but if your opponent knows how to counter it (just like scissors counter paper), then this strategy breaks down.


That doesn't really fit the definition of an optimal strategy however. An opponent with no prior experience but with an understanding of the rules and the ability to plan ahead would trivially be able to defeat this.

Most of the rest you talk about involves anticipating future actions and reactions, which is outside the scope of what I'm interested in, because although it is interesting it its own right, it really just ignores the issue by substituting temporal changes in choices for initial breadth of choice.

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1) Is an optimal strategy the best that the player can choose or the best that Laplace's Demon can choose? That is, is the optimal strategy determined by information available to the player or is it determined by the entire game state? Take the Monty Hall problem. If it's determined by the information the player has, then the optimal solution is to switch doors, but, if it's determined by the entire game state, then the optimal solution is to switch if the player chose the wrong door first and not switch if the player chose the correct door first.

2) If a strategy will not always win, can we still call it optimal? That is, is the optimal solution the one that is most likely to win, or the one that will win? This is similar to (1), but now I'm also asking if a game of chance like roullette where, even though you know the entire game state you still cannot determine the outcome (Laplace's Quantum Demon?), can have an optimal strategy.

3) If two strategies are equally likely to win, are they both considered optimal? Consider a coin flip. For simplicity, we'll ignore the possibility of it landing on its edge. If the coin is fair (50/50), then there are two equal strategies; are either, both, or neither of them considered optimal? If the coin is biased but the player doesn't know which way, does that change the answer?

4) If a strategy is only suboptimal by a small amount, does that matter? Let's say in the coin flip the play knows that the coin is biased 51/49. In this case, even chosing the suboptimal strategy is very likely to produce a win.

5) Can a strategy take into consideration the metagame? That is, can it bluff? Can it read other players? Can it count cards?

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1) I'm primarily thinking of complete information games, in the belief that partial information games can be considered an extension of that. However I think the distinction between a typical player and Laplace's demon (linked for the benefit of others) is very important, as there is probably an optimal strategy for chess, but a typical player can't find it. This demonstrates that it's possible to make games which are essentially unpredictable, yet which require no random factors to make that so.

2) Yes, the optimal strategy just needs to be the most rational one that you'd choose, if all the information was available to you. I'm mostly ignoring stochastic effects for now though, assuming that they average out over enough trials.

3) I'd say yes - if there is no strategy more likely to yield victory, then it's optimal, even if the utility of that strategy is equal to that of the worst strategy. Obviously a game with these properties wouldn't be much fun.

4) Sure, why wouldn't it?

5) I'm trying to avoid any metagaming aspects, especially iterative versions of a game (eg. iterative RPS, or iterative Prisoners' Dilemma), because I don't think they are relevant here - obviously some games rely on adapting to change as a gameplay feature (whether slowly as in Civilization, or quickly as in a beat-em-up), but ultimately it usually only adds a trivial extra layer - you're either obviously compelled to change your strategy in response, or not.


I don't want to get fixated on the definition of the optimal strategy as such though. The key thing I'm trying to get at, is that it should be possible to design a game that has enough depth to stop players from being able to see the best thing to do very easily, yet without relying on random opponent behaviour - whether that be by dice rolls or by human unpredictability - to create that depth.

I'm just trying to work out how that depth is created - interaction between multiple aspects seems to be the key here, as someone mentioned above, but making that interaction non-trivial is important too. I think it's interesting that most complex games seem to have several resources in play, and you can trade one for the other. One book I read described chess as having material (pieces you've taken), position (areas of the board you control), and tempo (how well you've made uise of the time available). Magic: The Gathering has your in-play cards, your hand, your deck, and your health score. Civilization has money, food/trade/science resources, cities, and units. The complex ways in which these resources affect each other and can be converted into each other might be a major part of how these games create their depth.

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