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Worlds of Design: Always Tell Me the Odds

If GMs (and game designers, and gamers) understand “the odds” they will be able to make better choices and understand why some things happen in their games - and some don’t.

Never tell me the odds!
--Han Solo (Star Wars)​

Most people don't understand odds and randomness in the most simple dimensions, especially when you're talking about dynamic odds.
--Keith S. Whyte. Executive Director. National Council on Problem Gambling​

We often hear about “the percentages” and “the odds” in sports. For example, the odds for the home team winning (regular season: NBA 59.9%, NFL 57.1, NHL 55.1, MLB 54.0, MLS soccer (where there are draws) home win ratio of 49.4 percent over a 15 year period, compared to just 26.5 percent away wins). Though game design does not require higher math, game designers need to know simple arithmetic and probability. There are some odds we can talk about in RPGs, as well, and about how people react to those odds.

The notion that it can be a "fair fight" in an RPG? 50/50? Nope.

How much is a fight biased toward the adventurers? Let’s consider the NCAA Basketball tournament. Let’s say that a team is so good, it can win 90% of its games against the better teams, the ones in the tournament. This is unlikely: how many teams have a season record as good as 27-3 (90%) though they’re playing weak as well as strong teams? When you’re playing the stronger teams, 90% is quite unlikely. But let’s use that anyway.

So what are the chances of winning the tournament (six games in a row) even with that 90% (beyond-likelihood) capability?

90%​
win 1 in a row​
81.00%​
win 2 in a row​
72.90%​
win 3 in a row​
65.61%​
win 4 in a row​
59.05%​
win 5 in a row​
53.14%​
win 6 in a row​

Even that most unlikely team that can win 90% of games against tournament-quality opposition, only has a 53.14% chance of winning the tournament. Even a team with a 99% win likelihood wins the six-game tournament only 94.15% of the time (“fail on a roll of 1 on d20").

(How is this calculated? You multiply, you don't add. So to win three games in a row, it’s 90% times 90% times 90%.)

This is why the “best team” often fails to win the tournament. This is why some pro sports play seven-game playoff series, in the hope that luck “evens out” and the better team will win.

Translate This into RPGs

Extrapolate that into RPG sessions with perhaps one big battle per session, or maybe more! Practically speaking, either you need really astute players willing to run away from almost any encounter, in order to avoid taking chances, or you need to arrange a huge bias in favor of the players in a typical encounter. Or they're going to lose and possibly die pretty soon.

Go back to the tournament example. If the players are 90% likely to win, after six encounters there will be around a 47% chance that they will have lost one of those encounters.

The whole notion of RPG combat as "sport", as something that's "fair", is nonsense in light of these calculations.

Playing Styles

Some play for "the rush", for glory, and like Han Solo don't want to know the odds before they do something. If you accommodate them, then the bias in favor of the players must be even greater, or you'll have dead characters in no time. (This brings up the question of "fudging" dice rolls in favor of characters, which I may address another time. Some GMs do it routinely, others never.)

Is it fun to play to survive, to “win”, instead of for glory? Depends on the person. It is for me, when I expand it to include survival for the entire group, not just my character(s). In contrast, in the late 70s I played in a game that was supposed to act as the stimulus for someone to write a story. I tried to do something "heroic". My character got dead.

Many gamers don't understand probability, and so over- (or under-) estimate their chances of success. Some don't understand the scope of the chances. 1 in a thousand vs 1 in a million is a massive difference, but people often don't see it that way. It's yet another case of perception not matching reality.

That's where we get those who don't understand odds, who think that anything (no matter how outlandish) ought to be possible once in 20 (a 20 on a d20) or at worst once in a hundred (100 on percentage dice). No, the chance of most anything happening in a given situation are astronomically against. (And "astronomically" is practically the same as "impossible".)

Recently I talked with a gamer who is very skeptical of probabilities, but doesn't understand them. He thought it was terribly unlikely that a player could roll five dice in a row and get at least a 4 on every roll. The chances, 50% to the fifth power, amount to better than 3%. For some reason he thought that rolling the dice successively rather than altogether made a difference - nope, what's come before has no bearing on what comes after, in odds. And what about five 1's in a row? That's 16.66% (a 1 on a d6) to the fifth, .000129 or .0129%. One tenth of one percent (one chance in a thousand) is .01%. So slightly better than one chance in a thousand. Rolling seven 1's in a row is about 3.5 chances in a million. Or perhaps more easily, rolling a 1 on every one of six 10-sided dice is a one-in-a-million chance.

To summarize: For designers, fudging the dice (or the quality of the opposition) is inevitable. For players, it helps to understand probabilities in games

Reference: James Ernest (Cheapass Games) - Probability for Game Designers | League of Gamemakers
 
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Lewis Pulsipher

Comments

CapnZapp

Legend
I think the math matters if the table decides it does. This is because D&D can be played in many, not necessarily disjoint ways, such as (wargame, story, TotM, ...).
If a game's math works, it can be ignored if you don't need it.

If it doesn't work, you can't just magically make it work if you do.

In other words, there's zero justification for broken game math. That doesn't mean you must play that way, only it needs to be there for those who want to use it.
 

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prabe

Aspiring Lurker (He/Him)
Supporter
If a game's math works, it can be ignored if you don't need it.

If it doesn't work, you can't just magically make it work if you do.

In other words, there's zero justification for broken game math. That doesn't mean you must play that way, only it needs to be there for those who want to use it.
Related: If the game's math is broken, and you're not a math person, you won't know why the game isn't working the way you expect and you won't have a prayer of fixing it.
 


Hussar

Legend
What I find magical is how you got from what I said to where you went. :sneaky:
Not speaking for anyone else, but, I went the exact same way when I read your post. Could you perhaps rephrase it because, apparently, I'm not the only one who missed what you meant.
 

CodeFlayer

Explorer
Sure - I'm in a thread discussing the appropriateness of teaching and/or requiring a working knowledge of probability to play reasonably well, as a wargame. What I am suggesting is that it is the body of players at the table together that decide what the ground rules are for the RPG. Most games I have seen give the GM wide latitude. It is the shared experience, and the quality of it, that is paramount. That means that, with the table's consent, I can take control of the mathematics of the simulation. The game designers don't reach that deep or that far, as I see it.

It is not my intend to sow discord - so please explain how you took my words some other way. I am new here.
 

Hussar

Legend
Ok. That clears it up. Your initial post made it sound more like math doesn't really matter because the group will just "work around" bad math.

Yes, we need to give the DM pretty wide latitude, fair enough. But, there does come a time when it should be appropriate, or, at least not seen as antagonistic, to question the DM's math. I find that in situations where the rules aren't terribly explicit, DM's often err far too much on the side of caution which turns a difficult task into one that's virtually impossible, or, where the rewards aren't worth the risk.

Take the old saw about swinging by the chandelier across the room to attack someone. There is a school of thought which says that you have to break that down into several individual actions in order to succeed. Jump to the chandelier, cut the rope holding it in place, make an attack. And, if any of those checks fail, the entire attack fails and the action is lost.

So, what sort of benefit should we give the PC for attempting something like this? Say it's a 50:50 chance for each step. That's a 1 in 8 chance of success. IOW, even though each step doesn't look that hard, (most DM's wouldn't consider a 50% chance of success as hard), by requiring so many checks, it becomes extremely unlikely to succeed. So, if you have a 1 in 8 chance of success, the reward has to be at least 4 times greater than if you just shot him with a ranged weapon.

How many DM's would allow you to deal 4X damage for this maneuver?

This is what I keep coming back to. Because the risk:reward calculation is so bad, no one tries doing anything outside the box because, most of the time, anything outside the box is either going to fail, or will never actually reward you as much as it should.

-----

Edited to fix math. Dammit. :p
 
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clearstream

Be just and fear not...
Ok. That clears it up. Your initial post made it sound more like math doesn't really matter because the group will just "work around" bad math.

Yes, we need to give the DM pretty wide latitude, fair enough. But, there does come a time when it should be appropriate, or, at least not seen as antagonistic, to question the DM's math. I find that in situations where the rules aren't terribly explicit, DM's often err far too much on the side of caution which turns a difficult task into one that's virtually impossible, or, where the rewards aren't worth the risk.

Take the old saw about swinging by the chandelier across the room to attack someone. There is a school of thought which says that you have to break that down into several individual actions in order to succeed. Jump to the chandelier, cut the rope holding it in place, make an attack. And, if any of those checks fail, the entire attack fails and the action is lost.

So, what sort of benefit should we give the PC for attempting something like this? Say it's a 50:50 chance for each step. That's a 1 in 8 chance of success. IOW, even though each step doesn't look that hard, (most DM's wouldn't consider a 50% chance of success as hard), by requiring so many checks, it becomes extremely unlikely to succeed. So, if you have a 1 in 8 chance of success, the reward has to be at least 7 times greater than if you just shot him with a ranged weapon.

How many DM's would allow you to deal 8X damage for this maneuver?

This is what I keep coming back to. Because the risk:reward calculation is so bad, no one tries doing anything outside the box because, most of the time, anything outside the box is either going to fail, or will never actually reward you as much as it should.
It's tricky, though. The multiple of damage need not be the same as the change in chance to hit: the two are related, but not perfectly commensurable. Consider the power attacks (-5 for +10 damage).

And then there is also the nuance that I was going to hit in one of two possible worlds, now one of eight, so I would need four (not eight) attempts to be back to where I was. Suggesting a 4x multiple. I'm not knocking your example, only trying to say that the implications are far from straightforward even where one knows the odds. In a d20 system it might be about right to give +1 to +2 damage for each -1 to hit. I went from 11+ to about 18+ so anything from +7 to +14 damage could be right.

I honestly believe the conversation is somewhat empty without factoring in stakes. And one way to think about the odds is as offers to players (so that a DM offers a reward at some cost at some odds). It's usually up to the player if they choose to accept that offer (or should be). In our swinging chandelier incident, the offer might rightly be something equating to a power attack.

I wonder what the role of the player ought to be? Is the DMs responsibility to never offer unfair odds? When might it be okay to do so? Is the DM always required to know the odds, or do players bear some responsibility for that?
 

Hussar

Legend
Heh. Wasn't quick enough with my edit. Yeah, that should be 4x damage, not 8x. :p I hate math.

If the DM is offering unfair odds, in a game, then, IMO, that should be something very rare. And, frankly, if the odds are unfair, then the player probably shouldn't do it.

But, yes, you are totally right that it's not a straight line relationship. It's far, far more complex than my example. Totally agree.
 

CapnZapp

Legend
Related: If the game's math is broken, and you're not a math person, you won't know why the game isn't working the way you expect and you won't have a prayer of fixing it.
Sure.

As long as you aren't saying this to argue games don't have to have sound math, okay. Edit: which does not have to be true, either for you or the other poster.
 

clearstream

Be just and fear not...
Heh. Wasn't quick enough with my edit. Yeah, that should be 4x damage, not 8x. :p I hate math.

If the DM is offering unfair odds, in a game, then, IMO, that should be something very rare. And, frankly, if the odds are unfair, then the player probably shouldn't do it.

But, yes, you are totally right that it's not a straight line relationship. It's far, far more complex than my example. Totally agree.
I'd argue that the odds should almost always be marginally unfair :p

But to your point, it seems like a DM should have a decent sense of what they are offering, and should not persistently offer unfair odds. What is unfair? That might be defined as when whatever is at stake against whatever is rewarded, multiplied by the odds, trends strongly towards lossiness. Players get much less out than they put in.

There is the matter of how obvious the risks are, but there I am suggesting that what is at issue there is not unfairness at all. The DM does not have sole responsibility for sizing or even necessarily elucidating risks. The riskiest risks are those that have obscured components!
 

prabe

Aspiring Lurker (He/Him)
Supporter
Sure.

As long as you aren't saying this to argue games don't have to have sound math, okay. Edit: which does not have to be true, either for you or the other poster.
The opposite of that. I agree with you that games need sound math. I was speaking as an English major who loves rules-tinkering and occasionally gets in over his head and/or runs headlong into unanticipated outcomes.
 

Fenris-77

Small God of the Dozens
Supporter
I think there are more ways of rewarding swinging from chandeliers than just additional damage. Assuming for a moment that you actually want your players doing heroic swashbuckling stuff like that in the first place, which admittedly may not be every campaign. Rewarding that sort of action with damage is .. um .. weird, to me anyway. But it should be rewarded, and in fact needs to be rewarded if I want to happen on a regular basis. My first instinct is to break the action down into less parts and generally be less punitive about overall chances of success. That allows me to scale down the reward. Second, I would probably grant rewards here that lie outside the RAW. I really like some version of Inspiration here, but buffed enough that it's a more tangible reward, maybe closer to the utility of a FATE point. Not everyone wants to add mechanics though. Another option is to link actions outside attack/damage with results that are also outside attack/damage. To keep the swashbuckling example, lets say you want to kick the Cardinal's guards down the stairs they just came up. That's a pretty standard swashbuckle right? If I dip into the damage rules to adjudicate that, I kind of need to stay inside the damage rules, which have pretty strict rules for things like knockback and prone and whatever. But, if the action is adjudicated as a simple cause/effect, for example, I want to swing from the chandelier and knock the guards down the stairs, then I can just give it a DC and roll with the narrative.

I guess what I'm getting at is that it's way easier to keep risk and reward balanced when you keep the attached mechanic simple. Lots of DMs don;t really think about the actual math behind asking for a series of checks though, and I think that does serve to keep PCs action choice very much inside the box.
 

prabe

Aspiring Lurker (He/Him)
Supporter
I think there are more ways of rewarding swinging from chandeliers than just additional damage. Assuming for a moment that you actually want your players doing heroic swashbuckling stuff like that in the first place, which admittedly may not be every campaign. Rewarding that sort of action with damage is .. um .. weird, to me anyway. But it should be rewarded, and in fact needs to be rewarded if I want to happen on a regular basis. My first instinct is to break the action down into less parts and generally be less punitive about overall chances of success. That allows me to scale down the reward. Second, I would probably grant rewards here that lie outside the RAW. I really like some version of Inspiration here, but buffed enough that it's a more tangible reward, maybe closer to the utility of a FATE point. Not everyone wants to add mechanics though. Another option is to link actions outside attack/damage with results that are also outside attack/damage. To keep the swashbuckling example, lets say you want to kick the Cardinal's guards down the stairs they just came up. That's a pretty standard swashbuckle right? If I dip into the damage rules to adjudicate that, I kind of need to stay inside the damage rules, which have pretty strict rules for things like knockback and prone and whatever. But, if the action is adjudicated as a simple cause/effect, for example, I want to swing from the chandelier and knock the guards down the stairs, then I can just give it a DC and roll with the narrative.

I guess what I'm getting at is that it's way easier to keep risk and reward balanced when you keep the attached mechanic simple. Lots of DMs don;t really think about the actual math behind asking for a series of checks though, and I think that does serve to keep PCs action choice very much inside the box.
If it's being used as an attack, I'm thinking something like Advantage on that attack, and probably something to reflect severely wrong-footing the enemy like the enemy are surprised for a round. In the instance of kicking them down the stairs ... Advantage on the attacker's roll to shove, Disadvantage on the defenders' rolls to resist (and allow the attacker to affect more than one enemy at once--it's cinematic!). There's plenty of flexibility in the system for stuff like that, if the DM can think at off-angles. It's a little harder for some (especially players), because there's so little explicit player-facing text to let them get a decent picture of their chances.
 

Fenris-77

Small God of the Dozens
Supporter
If it's being used as an attack, I'm thinking something like Advantage on that attack, and probably something to reflect severely wrong-footing the enemy like the enemy are surprised for a round. In the instance of kicking them down the stairs ... Advantage on the attacker's roll to shove, Disadvantage on the defenders' rolls to resist (and allow the attacker to affect more than one enemy at once--it's cinematic!). There's plenty of flexibility in the system for stuff like that, if the DM can think at off-angles. It's a little harder for some (especially players), because there's so little explicit player-facing text to let them get a decent picture of their chances.
I tend to use the, admittedly somewhat arbitrary, break point of 'does it do damage'. If all you're doing is kicking the guards down the stairs, i.e. they aren't taking HP damage, then I feel well justified adjudicating it outside the combat rules. You certainly can adjudicate it inside the combat rules and still have it be cinematic though, for sure. You'll notice your example also escapes the three rolls to succeed model presented above? That's really the key. Make it one roll, however you're doing it, and it becomes a more approachable idea, and the math is less likely to get on top of you.

Something that can be an important decision point here for the DM is the roll for initiative. Unlike the rest of the game, D&D has that very specific indicator that the game is NOW IN COMBAT MODE. Dun-dun-daaah!! Once you roll initiative expectations change a little. The stair kicking example is one I might allow as an action in response to my stating two guards appear at the top of the stairs. An action to delay or obstruct those guards doesn't have to happen after initiative is rolled. The stair kick, or throwing a table, or pulling down a tapestry, all those things can be resolved outside combat without breaking the game. That's part of why damage is my break point. It's really about the goal of the action. If the goal is to kick the guards down the stairs so that your escape attempt can continue, then rolling initiative just gums things up. I'd allow the action, and if it failed, then we're in combat. However, if the goal of kicking the guards down the stairs is to injure them, then it seems like a roll initiative and enter combat kind of idea.

I think your last sentence is really important. The players need to have some idea of how their swashbuckle may work, or they simply won't attempt it, or even suggest it. I think that the production of some player facing text might be necessary. Even a discussion in session zero about that sort of thing, where the DM sets some limits and manages expectations, could help. I prefer something more concrete though.
 

prabe

Aspiring Lurker (He/Him)
Supporter
I tend to use the, admittedly somewhat arbitrary, break point of 'does it do damage'. If all you're doing is kicking the guards down the stairs, i.e. they aren't taking HP damage, then I feel well justified adjudicating it outside the combat rules. You certainly can adjudicate it inside the combat rules and still have it be cinematic though, for sure. You'll notice your example also escapes the three rolls to succeed model presented above? That's really the key. Make it one roll, however you're doing it, and it becomes a more approachable idea, and the math is less likely to get on top of you.
To be honest, I was thinking a roll to Do the Chandelier Thing (probably the player's choice of STR or DEX, Athletics or Acrobatics), with the upside as described and the downside of ... falling damage? clinging to the chandelier as it swings 20' over the floor and the people on the floor start loading crossbows? dunno. It's still (really) one roll for the stunt itself modifying other rolls, but I concur that it's not so many as to ensure failure.

I think your last sentence is really important. The players need to have some idea of how their swashbuckle may work, or they simply won't attempt it, or even suggest it. I think that the production of some player facing text might be necessary. Even a discussion in session zero about that sort of thing, where the DM sets some limits and manages expectations, could help. I prefer something more concrete though.
Yeah. The players should know if the GM intends to reward that sort of creativity. I don't think that way much as a player, but I'd like to think I'm willing to work with a player who does.
 

clearstream

Be just and fear not...
If it's being used as an attack, I'm thinking something like Advantage on that attack, and probably something to reflect severely wrong-footing the enemy like the enemy are surprised for a round. In the instance of kicking them down the stairs ... Advantage on the attacker's roll to shove, Disadvantage on the defenders' rolls to resist (and allow the attacker to affect more than one enemy at once--it's cinematic!). There's plenty of flexibility in the system for stuff like that, if the DM can think at off-angles. It's a little harder for some (especially players), because there's so little explicit player-facing text to let them get a decent picture of their chances.
Well, I'm pretty sure @Hussar was not heavily committed to the example, but FWIW I like this more than damage and I think it points to how one can always simplify an ad-hoc mechanic. Grant a benefit and perhaps call for an acrobatics roll to land safely, or let the attack or special melee roll itself speak to that.
 

Hussar

Legend
Yes, it's not good to dive too deeply into the example. I was just illustrating, not trying to create an axiom.

And, yes, I think the point is very well made - the reward has to equate with the chance of failure. Note, that's not exactly the same as risk because risk can also entail adding additional effects on failure, such as "land safely".

But, again, if we're now adding negative consequences in addition to increased chances of failure, then the reward also needs to be increased as well. Say, sure, it's only one Acrobatics check to swing from the chandelier, but, if you fail, you land prone, then how much, in addition to allowing the attack, do we have to reward the player in order to entice them to actually do it?

Again, if the risk:reward calculation is worse than simply engaging in rules sanctioned actions, then a rational player won't do it. It's a bad idea for the player to do it. This is not an easy calculation to do.
 

billd91

Hobbit on Quest
Again, if the risk:reward calculation is worse than simply engaging in rules sanctioned actions, then a rational player won't do it. It's a bad idea for the player to do it. This is not an easy calculation to do.
As long as you’re not stacking on lots of chances for it to fail by requiring too many skill checks, I’m not sure the risk:reward calculation needs to be too big a deal, particularly if you think relatively small and simple. Wanna swing from the chandelier to attack? Give me a DC 15 Dex (Acrobatics) check - if you succeed, you get advantage on the attack (which for a rogue implies SO much more), fail and you attack with disadvantage.
Life‘s easier if you just keep things simple.
 

prabe

Aspiring Lurker (He/Him)
Supporter
As long as you’re not stacking on lots of chances for it to fail by requiring too many skill checks, I’m not sure the risk:reward calculation needs to be too big a deal, particularly if you think relatively small and simple. Wanna swing from the chandelier to attack? Give me a DC 15 Dex (Acrobatics) check - if you succeed, you get advantage on the attack (which for a rogue implies SO much more), fail and you attack with disadvantage.
Life‘s easier if you just keep things simple.
I might let someone make a STR (Athletics) check to accomplish the same aim--but otherwise I agree.
 

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