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Balancing in push your luck games

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jambojazz34
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Consider a game where the main mechanic is "push your luck" and you can bust and lose your turn (e.g. Port Royal).

I'm facing the impact of busting on the final scoring.

How can you balance what you gain by pushing and what you lose by busting?
Is it self-balanced and just up to the player knowing the probabilities and take risks, so that if he busts it's just bad play?
Or maybe is it self-unbalanced? :D
Is there anything the designer must take into account in that regard? Maybe total number of turns to lower the effects of each single busting? Some players could even bust every single turn... Is there anything you can do?

All your suggestion are well appreciated.

X3M
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A math perspective

Look at it like this:

A chance to win/loose is 50%.
The next step also has this chance, 50%.

The total chance of winning both steps is only 25%. Players consider this to be very low, to go all the way with push your luck.

You need a higher chance for winning. Or multiple steps for the player to walk on while the chances are high.

***

You need a percentage per batch, and see if the end percentage is somewhere around what you want. If it is 50% per player, for the whole game combined. You need to look at the number of push your luck moments.

If this is 3. Simply third root the 0,5. This will end up with 80% chance of winning per push your luck.
Know that the bust is 20% for each moment. But if players keep trying to push their luck. The total game will have a 50% again.
20% + 0,8*20% + 0,64*20% = 49%

Now, I don't know how many moments you have that count as push your luck moments.
I also don't know what your end goal winning percentage should be for all moments to be successes.
Nor do I know if you want each moment to be equal. Or that you want something along the lines of 90%*80%*70% (=50%).

But I do know that if you have push your luck moments and players need to compete with these against each other. The game needs something to encourage players to take the gamble if they feel they fall behind.

What is the penalty? Does the player loose 50% or so?
What does the player win? The best balances that I know, ask for an equal win reward as what the player may loose.
+50% or -50%
+100% or -100%
There is no +100% and -50%. Unless you want players to make a steady progress. But then, they will always take the gamble.

If you have questions, please ask.
If this post of mine was a complete crap in your eye's, tell me too. :)

jambojazz34
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basic rules

Honestly, and very sadly, i know probability math only on a base level.
I also lack in english language... what do you mean with "push your luck moment"?

That said, in my game you have half deck completely safe (say colored white), and half deck equally divided in, say, blue and red. You bust when you draw 2 blue cards, or 2 red cards.
The first drawn is always safe. If the first is a white, the second is always safe. If the first is a blue, you have 25% of busting on the second. If the second is a red, you have 50% of busting on the third.
You can never go under 50%, if we consider the deck so big that taking cards out will not affect probabilities, because white cards are 50% of the total.

These numbers are valid only at the beginning of the game, because players have the opportunities to buy cards that let them bust at 3 instead of only 2 on some colour, and so on.

When you bust, at the beginning of the game, you completely lose your progress on that turn and discard all the cards drawn. This could change if you buy cards that let you save something when you bust.

When you win, you gain some of the cards as resources, dependently on what other players have done on your turn (they could prevent you from winning every card you drawn). Generally, most of the safe cards should be blocked by the other players, and most of the risk cards should not. So the player should statistically win about 1-2 cards at the beginning of the game if he doesn't get very lucky (players have limited cards for blocking) and other players play against him.

I can't say if that's an equal win/loss reward... maybe you can

X3M
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English isn't my primary language either

Don't worry about not knowing math. You guessed perfectly right on the number of cards that a player will pick.

The moment of push your luck. Well, you have described your game as one big moment. That is ok.

There are games out there that require the player to plan ahead their push of luck by having multiple smaller moments with great risk. Although the chance to loose per moment is smaller. The total chance of losing it all would be considerate bigger.

War games are a very fine example of this.

***

Regarding what you have described.

If white cards are safe, and you are allowed to pick one other card with a colour. The number of cards that a player can pick while taking no risk at all.
1: 100%
2: 50%
3: 25%
4: 12,5%
5: 6,25%

While chances are halved each time. It is better to look at the average number of cards that a player will be drawing in a long run. It is 2 on average for an infinite deck. 1-2 cards would be if you only consider the result on the first draw.

A little warning, I have not yet looked at having either a blue or red card. It will be a little higher then the average of 2.

Because I think that psychologically, players will stop drawing cards, once they have drawn either a blue or red card. The chances are just too high to bust. Especially with what you have gained so far.

If the risk is 25% to bust. Players could risk this at a maximum of 3 cards. Just to gain a 4th. But having 4 or more cards. And having that chance to bust. Well, the reward is not worth the risk.

The bust will be 25% chance to loose 100%. That is -25% in math terms. The reward needs to be bigger on average for an attempt.

The reward on 2 cards is 50%.
The reward on 3 cards is 33%.
The reward on 4 cards is 25%. And that is the turning point. I would not take the risk here to get a 5th card at all.

If you start at 2 cards and want to go to 4 cards. Then the rewards are 50% * 33% is 16,7%. This too will be considered to low for some players. When they had success with their first probable of 50% reward bust draw. They will not risk that 33% reward bust draw.

Players will be drawing at least 1 card.
2 is the average.
4 is the psychological maximum.

I hope this information has some use.

Fri
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My push your luck game opion and some probabilities

My opinion:

In my opinion a good push your luck game reward and risk need to increase during the duration of the turn. At the beginning of the turn the risk is low
but so are the rewards, most players will keep going. The middle area has more moderate risk but greater reward. Players should be rewarded for stopping here, but not too much. If players choose to keep going the should have a smaller chance of larger rewards.

Increasing the iterations will limit the effects of being unlucky.

*wrote the above before you posted more info*

Some probabilities:

Did some quick programming to come up with the chance of failure. That is the chance of drawing 2 of red or 2 blue color after drawing 3, 4 and 5 cards out of a 40 card deck that meets your description. (After 5 cards the simple programming I'm using takes to long to complete.)

3-29.75%
4-49%
5-65%

The chances of failure are even greater than this if players are holding holding white cards out of the deck.

It seems to me (though I haven't done anything confirm this) that after 5 cards its likely that a red and blue will be out, so as you guys have already figured out after 5 you probably have a 50% chance of failure on each turn after 5.

If your game is not working out now I would recommend trying increase the rewards or decrease risk or both.

Bonus humor:
English is my native language, but I am awful at applying it in written form.

let-off studios
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Incan Gold

TL;DR: You can self-balance by incorporating simultaneous turns so everyone faces the same decision at the same time; reduction of hazards the more frequently hazards affect the players; and increasing rewards for those that choose to risk by pushing their luck longer/more frequently.

More Details:
A game I absolutely love is Incan Gold, and it relies heavily on push-your-luck and self-balancing. It also uses the "draw one hazard card and it's a warning, draw a second of the same hazard card and you're out" mechanic, similar to what you describe in your original post.

It self-balances because the deck is reshuffled at the end of every round, thusly:
- the fewer players still in on a round, the more their potential reward. Those who continue to push will earn greater rewards.
- if players all stop pushing before a pair of matching hazards is drawn, then the round ends normally.
- if one player drops out before the others, the final player that successfully stops pushing collects all remaining treasures. In other words, you're rewarded more for pushing your luck more.
- if all players keep pushing until a pair of matching hazards makes them lose everything, one of that pair of hazards is removed from the deck before reshuffling and starting the next round.

There are also some other dynamics at play, including unique treasures that are won when only a single player pulls out. It's a fascinating game that does a lot with the push-your-luck mechanics, and I thoroughly enjoy playing it with large groups (up to 8 players).

See it here, should you be interested:
https://boardgamegeek.com/boardgame/37759/diamant

Never played Port Royal, if I recall correctly, so unfortunately I can't comment on that here.

jambojazz34
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Port Royal

I know Incan Gold, similar to Celestia, where the path you push your luck in is the same for every player. Very smart mechanics.

My game is more similar to Port Royal though.

In PR on a player turn, that player keeps drawing one card at a time (the other players just watch, with some exceptions for special powers). Some cards are colored pirates ships, that can make you earn golds, some cards are characters that you can buy with golds and give you special powers and victory points.
Characters never make you bust, but don't make you earn money.
Ships are the opposite. If you draw 2 ships of the same colour (there are about 5 colours), you bust and lose everything. If you stop before 4 different colored ships are shown without busting, you can pick 1 card of the row and get the money written on it (if it's a ship) or buy it (if it's a character). But if you stop at 4 different colored ships you can pick 2 cards, and if you stop at 5 (all the colours) you can pick 3 cards.

The probability of drawing 5 different cards in a row should be 1*0,8*0,6*0,4*0,2 = 3,84%... correct me if i'm wrong.

Ships have 1 to 4 golds (the 1 ones are the most common, the 4 ones are very rare).

I want to analyze the "the more push, the more you gain" aspect of that game.

Does this system incourage you to push your luck? It seems you should bust 96 time out of 100, to draw the 5 different colours just 4 times (and gain like 4times*3cards=12 turns-worth of gold)

I'm not considering special powers yet, that you don't own at the beginning of the game.

And the real question is: is this game still balanced?

I want to bring an example of gameplay.
Player A draws a card, it's a 4 gold worth of ship. He stops and gain 4 gold.
Player B draws a card, it's a 1 gold ship. He draws again, it's the same colour ship, he busts and loses the turn.

Player B pressed his luck when all the chances where in his favour, and still came behind with no apparent fault. Is this a matter of luck or a matter of balancing?

I have a hard time telling the difference...

jambojazz34
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Some questions

It seems to me that about 4 cards as psychological maximun is a nice spot.

Do you think rewards are too low? Does the bust/reward ratio affect only the number of cards player will on average draw or is it a matter of balancing?

How can you increase a reward? Is it only about lowering the probabilities of busting?

If that probability is the same for everybody, is the game balanced?

let-off studios
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Responses

jambojazz34 wrote:
How can you increase a reward? Is it only about lowering the probabilities of busting?

If that probability is the same for everybody, is the game balanced?

I watched the Dice Tower review of Port Royal. Wow! That game looks very enjoyable to me personally, and I definitely see why you'd consider it as an inspiration.

Regarding your questions:

INCREASING REWARD
I don't think you should increase the reward by reducing the chance of a player going bust. It sounds to me like that sets up a feedback loop where a player will simply bust a few times early in the game, but later they'll just keep pushing and pushing without worrying about going bust. Their turn becomes longer and the potential of a runaway leader is higher.

Port Royal rewards pushing your luck by:
- increasing available options (more cards to choose from at the end of a turn)
- increasing amount of cards one may collect (if you've drawn more cards, you can collect more before any other players have a chance)
- increasing options for other players to purchase, and therefore reward the player that drew them (if non-active players purchase a card, they pay the active player 1 gold)

IS IT BALANCED?
The more a game relies on chance, the less fair it can be. In a push-your-luck game, it seems to me that the fun of the game is experiencing a "lucky break" where you run away with great gains after great risk. Additionally, some players enjoy seeing just how far they can push it until they finally bust, and that "Awww!" moment makes for a memorable experience.

In both of these situations, the notion that it's a light game is a significant factor. Relying on luck both keeps analysis paralysis lower, as well as makes a loser feel better and a winner more gracious. There can be big swings between the winners and losers (which is why, for example, Incan Gold conceals collected winnings beneath the tents). But as it's a light game, it's not such a big deal.

CONSOLATION
To address your example: if you feel you need to build in a safety for those who seem to be encountering a bad luck streak, then build in a consolation prize. For example, in Port Royal I can imagine a crew member that one can hire that allows a player to draw a card even if they bust. Or what if a player collects a one-use flag token that matches the two ship cards that caused them to bust... This token can be used on a future turn to instantly discard one ship that matches the flag token's colour.

Consolation is useful in a game that has individual player turns as opposed to simultaneous turns, in my experience. So if you're designing something that is more like Port Royal and less like Incan Gold, it may be worth a try.

X3M
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jambojazz34 wrote:It seems to

jambojazz34 wrote:
It seems to me that about 4 cards as psychological maximun is a nice spot.

Do you think rewards are too low? Does the bust/reward ratio affect only the number of cards player will on average draw or is it a matter of balancing?

How can you increase a reward? Is it only about lowering the probabilities of busting?

If that probability is the same for everybody, is the game balanced?


I don't have much time right now, so I keep it short.

If those 4 cards are a nice spot.
Then the rewards are just right.
If you want to increase the rewards. You need to lower the chances of busting. But the 4 cards will be higher as well. If the cards are the rewards, then this can be your goal. You can increase chances by reducing the number of red and blue cards. Or, you bust a player when it has 3 red or 3 blue cards.

If everyone has the exact same chances, the game is balanced. However, a deck is limited, not infinite. Especially if one player picks all its cards. And a lot of white cards are picked. Then the next player has less chance having success. This will balance itself as well, since that player knows his/her chances are lower. But only that would be the unfair part. Maybe you can have all players pick 1 card before they pick again. That way, the chances are somewhat more leveled out.

jambojazz34
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Thank you

Thanks everybody, you helped me a lot!

cheers

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