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Way of the Dragon» Forums » Variants

Subject: Extra turn after equilibrium instead of after perfection rss

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Stephan
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In my experience, players try to go for multiples of the same symbol on every turn. This means that "perfection" (5 of a kind) is the best outcome of a normal turn.

Trying to go for "equilibrium" (1 of each symbol) is risky because you can usually only move 1 or 2 spaces if it doesn't work. And even if it does work, you can often move only 3 or 4 pieces because the others are blocked or already "home".

To balance this risk, I think it would be more fun to award the extra turn after a player rolls equilibrium, not after they roll perfection.
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Néstor Romeral Andrés
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Celtic Joker wrote:
In my experience, players try to go for multiples of the same symbol on every turn. This means that "perfection" (5 of a kind) is the best outcome of a normal turn.

Trying to go for "equilibrium" (1 of each symbol) is risky because you can usually only move 1 or 2 spaces if it doesn't work. And even if it does work, you can often move only 3 or 4 pieces because the others are blocked or already "home".

To balance this risk, I think it would be more fun to award the extra turn after a player rolls equilibrium, not after they roll perfection.


Good one. Please try and let us know how it goes. My only concern is that Equilibrium is way easier to achieve than Perfection.
 
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Stephan
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Mathematically it is easier to achieve (I think), but in practice hardly anyone tries to achieve it because of the high risk of a bad turn and the rather low reward of 3, maybe 4, movement (in my experience).
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Nathan Morse
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If I'm one symbol off, and especially if I would get 5 spaces for it, I often will chance equilibrium. Also, equilibrium is an uncommon and delightful way to start a turn with the initial roll, but I've seen it happen often enough. Starting a turn with perfection… well, let's look at the statistics:
equilibrium: 5! ⅙⁵ = 120 combinations out of the potential 7776 for an opening roll
perfection: 5 ⅙⁵ = 5 combinations out of the potential 7776 for an opening roll
So, equilibrium is 24× as likely as perfection for an opening roll (which is consistent with my experience). Similarly, being only a 1-in-6 roll away from either of them on your opening roll, unless I'm mistaken, is as follows:
one-short-of-equilibrium: 600 out of 7776
one-short-of-perfection: 25 out of 7776

That said, I think your risk-balancing argument is interesting. Assume for a moment that you only get a single roll, and that we are only looking at the first move of the game, with no obstructions to kill your options. Equilibrium is 24× as likely, but being the typical one roll short means you move 2 spaces (as you said). Perfection is 1/24 as likely, but being one roll short moves you 4 spaces. Let's extrapolate this a moment, and see where it goes.

Filtering all other results besides nearly or completely successful attempts at these special rolls… Of course, 1 in 6 attempts to complete a one-short equilibrium or perfection succeed. The ratios of rolling their one-short forms are the same as their perfect forms. So, let's shrink our window to focus on the natural complete rolls.
For every 24 equilibria I roll (5 spaces), I roll 1 perfection (5 spaces + 1–5 + …).
That means for every 120 spaces I earn through equilibrium, for perfection I earn 5 + 1–5 + a 5/7776 chance at another 1–5 and so on — let's be generous and call it 10 spaces. OK, this is the easy part: Equilibrium clearly earns me like 12× as many spaces as perfection. No need to enhance it.

But now let's look at those painful endings of being one short.
For every 24 one-short equilibria I roll (2 spaces), I roll 1 one-short perfection (4 spaces).
That's 48 spaces of barely-failed equilibrium to 4 spaces of barely-failed perfection. That's still a factor of 12 in favor of equilibrium.

I grant that this is only two points of the set, and that I may have made mistakes, but this suggests to me that the 24×-as-high likelihood of equilibrium opportunities will yield some 12× as many spaces.

If you want to talk about rewarding risk, let's shift this discussion from equilibrium to calling the dragon! …especially because not only does calling the dragon yield only 1 space upon failure, even on success it yields 0 spaces if it's 4 dragons + a path you haven't started / a path on which you are in the lead (assuming you don't take the negative progress of swapping anyway). Even a successful success can earn you as little as 1 space, understanding that your setting an opponent back by the same amount (in a 2-player game, that's effectively a 2-space gain, but in a 5-player game, that's effectively more like a 1¼-space gain). On the other hand, it could be +11 spaces (and +5 points) for you and −11 spaces (and −5 points) for someone else in the incredibly unlikely opposite extreme.

All of these things are, to me, what define Way of the Dragon. I'm with Néstor: Try your variant for a good number of plays, and keep us posted!

P.S. I forgot to walk through the numbers for your variant: For every 1 perfection (5 spaces), we'd see 24 equilibria (5 spaces + 1–5 + …).
That means for every 5 spaces I earn through perfection, for equilibria I earn 24× (5 + 1–5 + a 120/7776 chance at another 1–5 and so on — again, let's be generous and call it 10 spaces, even though the chances for repeat turns is 24× as likely as with the normal game's perfection). So now I'm getting 5 spaces of perfection for every 240 or so spaces of equilibrium. The swapped bonuses cause equilibrium to give a good 48× as many spaces as perfection, instead of just 12× as many, so it definitely shifts the balance the direction you intended, but I'm not convinced this would be for the better. Still, let us know what you find, in practice, because the players' choices muck up all the easy statistics for this game (which is what makes it a good game)!
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Néstor Romeral Andrés
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That was a long post! Thanks!
 
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Stephan
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That was indeed interesting, Nathan.

However, in my experience Perfection and Equilibrium happen almost equally often in games, maybe 3 to 2 in favor of Equilibrium, but not more.

I think, in practice you can't calculate 5, or even 4 moves for Equilibrium, at least not in 3-player games, much less with 4 or 5.

And that's not even thinking about the second half of the game, where at least 2 of your pieces are either already home or 1 space away, in which case you really do not want to roll Equilibrium.
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Nathan Morse
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Absolutely correct about the actual "during the game" rolls. I usually avoid equilibrium if it can't give me at least 3 spaces.

So, to your point, I think people just start avoiding the 24× as likely option as it becomes a poorer payoff.

Still, looking forward to your "real life" empirical findings!
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Stephan
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I'm actually about to trade the game away (sorry, Nestor), but new ones are coming in, and old ones need to go.

So you won't get any test results from me. Sorry.
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Néstor Romeral Andrés
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Celtic Joker wrote:
I'm actually about to trade the game away (sorry, Nestor), but new ones are coming in, and old ones need to go.

So you won't get any test results from me. Sorry.


Someone else will enjoy it!
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Nathan Morse
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Celtic Joker wrote:
I'm actually about to trade the game away (sorry, Nestor), but new ones are coming in, and old ones need to go.

So you won't get any test results from me. Sorry.
Oh, I see, smacking the hornet's nest and then running, eh?

Enjoy the new games, Stephan!
 
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Grant Fikes
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One suggestion to make equilibrium more appealing as the game progresses is to have all pieces advance to the next empty space.
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