TLDR: Does anyone have/know of any good/easy *betting* rules for spectators of arena-based combat?
Really Long; Still Want to Read:
The Gladitorial Arena is a common and enjoyable trope for all fantasy games, and adjudicating its combats is fairly easy with knowledge of the combat system.
Adjudicating the betting system is not so easy, especially for someone like me that has fairly little knowledge of real world gambling.
I'm looking for some rules, preferably simplistic, but I can cope with complicated if necessary, on PCs placing bets on gladiator matches. Has anyone seen rules like this in a source? Does anyone with gambling knowledge want to try their hand at it? I would like to see something that DOESN'T involve a skill check, ideally. Simply having the PCs roll a gather information check takes all the risk out of it. Ideally, I'd like for it to be as realistic as possible ("I put 200 gold on Gorzax the Destroyer to down The Ripper in the second 30 seconds of the match, at 2:1 odds!", etc).
I'd be inclined to go as follows:
Start by setting (in your head) an actual advantage that Gorzax has (e.g. Gorzax is actually 3 times as likely to win as the Ripper, meaning betting odds of 3:1 are "fair.") Don't tell the players this, though.
Use "gather information" to find either a tip from the stables about who's going to win, or alternatively, to find the best odds (Jimmy the Greek is offering 5/2 on Gorzax, but Ludwig the Mad is offering 7/2, so we'll bet with Ludwig).
Everyone involved rolls as many dice as the true odds. (I'd roll d100s to minimize ties.) In the examples above, Gorzax rolls three dice, Ripper rolls one. Highest roll wins. (This means that Gorzax does, in fact, have exactly three times as many chances to win. Ludwig probably loses money on the fight. That's why he's MAAAAAAAAD!!!!1!!11!!!eleven!!)
You can either dice off ties or simply say that the tie is a split decision and all bets are off.
I think this system would be fast and playable. The problem with too much realism is you'd basically have to have an arena-simulation system and someone very knowledgeable about statistics to be able to estimate the odds that the professional gamblers would be offering.
The advantage here is that skill rolls don't take all the risk out of it; they just let you correctly assess the risk.
If you like, you can then play out the combat using combat rules, but fudging dice rolls to make sure that the right person wins. This would work well if the PCs are only spectators.
If, as is also a common trope, the Ripper is actually a PC,.... how are the professional gamblers going to be able to size up his form? About the best you can do is estimate CR vs CR (which presumably they can) and then fight it "fairly." I would guess quickly that each CR is one "level" of odds; e.g. if the Ripper is a CR 6 PC going against a CR 6 opponent (or 2 CR 4 opponents, effective CR 6), then the odds are fair (1:1). A CR 6 PC against a CR 8 opponent would be 3:1 in favor of the opponent....
Odds ratios are inverted fractions. 2:1 means there's a 50% chance that the given fighter will win, and thus money will pay out double if you do win.
Figure out the approximate fraction chance that a given outcome will happen, flip it, those are the odds. Remember that bets with high numbers are extremely unlikely, so having the underdog always win just means your PCs will make out like bandits. Have the easy money win more but pay out way less, and the low chance numbers rarely work but work sometimes, and of course pay out high.
Don't let narratives ruin statistics, or your PCs will get free money off plot guesses!
Odds ratios are inverted fractions. 2:1 means there's a 50% chance that the given fighter will win, and thus money will pay out double if you do win.
Er... 2:1 means that the stronger fighter is expected to win about 2/3 of the time, and if the weaker fighter wins, you get your stake returned, plus double what you bet as winnings.
A "fair" bet (aka evens) is 1:1.
You can see that by looking at a roulette wheel. Odd/even (or red/black) pays "even money 1:1", while columns pays 2:1 (your stake plus twice your stake).
ETA: Formally, if the odds are X:Y, this means that you're expected to win your bet about Y times for every X times you lose. This means that the actual probability of winning is Y / (X+Y), for those that really groove on algebra.
This is why, if the odds are 4:1, the favorite rolls four dice and the underdog rolls once. With fair dice, the winning number is equally likely to appear on any die, hence the underdog will win one time for every four times the favorite wins.
If you're willing to roll fistfuls of dice, you can simulate very nuanced odds. If the odds are 11:9, the chances of the underdog winning is about 45%.
Parimutuel betting is probably the simplest form of betting to use for this situation (and fairly fitting). The odds are calculated simply by how much money is thrown into each fighter. More money means better odds, but lower payout. You don't have to complicate it with time limits and such. Just keep it simple.
You could set up X amount to be given to each fighter, then maybe allow people to use Diplomacy/Bluff checks to "play-up" or "downplay" your fighter (ie "Man, I heard Gorzax sprained his ankle earlier today. Hope he's still able to fight...). That would maybe change the payout in a predetermined way, or just move 100 gp from one person to the other or whatnot.
Parimutuel betting is probably the simplest form of betting to use for this situation (and fairly fitting). The odds are calculated simply by how much money is thrown into each fighter. More money means better odds, but lower payout. You don't have to complicate it with time limits and such. Just keep it simple.
You could set up X amount to be given to each fighter, then maybe allow people to use Diplomacy/Bluff checks to "play-up" or "downplay" your fighter (ie "Man, I heard Gorzax sprained his ankle earlier today. Hope he's still able to fight...). That would maybe change the payout in a predetermined way, or just move 100 gp from one person to the other or whatnot.
The problem with this is that it doesn't provide any way of representing the intelligence and skill of the regular gamblers. The guys who have been coming down to the pits every Starday for the past five years have a pretty good idea of how good the regular fighters are, and they're not going to be dumping their money evenly (or randomly) between the two. If Gorzax hasn't lost a bout in his past twenty, people are going to be pouring their money into him.
For a real life example, look at the NFL games coming up. I wish I could get a bet down on the Denver Broncos at straight up. Instead, the "smart" money decided several weeks ago that the Broncos were a better team, possibly because they've been playing much better than their opponent. Sure, I could try to start a rumor on the internet that the Denver quarterback has just sprained his ankle, and that might move some money, but I'm starting at a huge (and justified) disadvantage.....
Parimutuel betting is probably the simplest form of betting to use for this situation (and fairly fitting). The odds are calculated simply by how much money is thrown into each fighter. More money means better odds, but lower payout. You don't have to complicate it with time limits and such. Just keep it simple.
You could set up X amount to be given to each fighter, then maybe allow people to use Diplomacy/Bluff checks to "play-up" or "downplay" your fighter (ie "Man, I heard Gorzax sprained his ankle earlier today. Hope he's still able to fight...). That would maybe change the payout in a predetermined way, or just move 100 gp from one person to the other or whatnot.
The problem with this is that it doesn't provide any way of representing the intelligence and skill of the regular gamblers. The guys who have been coming down to the pits every Starday for the past five years have a pretty good idea of how good the regular fighters are, and they're not going to be dumping their money evenly (or randomly) between the two. If Gorzax hasn't lost a bout in his past twenty, people are going to be pouring their money into him.
For a real life example, look at the NFL games coming up. I wish I could get a bet down on the Denver Broncos at straight up. Instead, the "smart" money decided several weeks ago that the Broncos were a better team, possibly because they've been playing much better than their opponent. Sure, I could try to start a rumor on the internet that the Denver quarterback has just sprained his ankle, and that might move some money, but I'm starting at a huge (and justified) disadvantage.....
To be fair, he is looking for something fairly simplistic. And it does take into account the history of a fighter, since they would be getting more of the money in bets. The playing up/downplaying idea was just something of an idea. Besides, gamblers are a superstitious lot. I'm sure an enterprising bard or rogue could play on that a bit :)
A good example, you've got Gorzax the Destroyer vs Timmy the Eel. Timmy the Eel is an underdog and still green, while Gorzax is a very seasoned veteran. When the bets are thrown in and closed, a total of, say, 5,000gp was bet on Gorzax, while Timmy the Eel only got 100gp. The grand total of the pool is 5,100 gp. From here, you can calculate the odds.
The odds of Timmy the Eel defeating Gorzax the Destroyer:
5100gp / 100gp = 51 gp per 1 gp wagered.
Essentially a 50 gp profit per 1 gp spent, so it comes to the odds of 50:1 that he wins, or 1.9% ({1/[50+1]}*100) chance he wins. Very low odds, but the pay out will be pretty awesome if he wins. If I put in 10 gp, then I'd get that 10 gp back as well as 500 gp profit. And obviously, the losers get diddly squat.
The odds of Gorzax the Destroyer defeating Timmy the Eel:
5100gp / 5000gp = 1.02 gp per 1 gp wagered.
So for this, it's only a .02 g (2cp) profit per 1g wagered. The odds here (.2:1, though for ease of reading, most would make this a whole number, so 1:5) would become the percentage 98%.
This is, of course, assuming the arena isn't taking out a percentage cut. I kept this simple for you, but you are more than welcome to take off 14% or so from the pool before giving it out to everyone that won. I hope this helps.
NOTE:
Formula for turning odds (ie 5-1, 5:2) into probability is:
Odds (a:b)
Probability = (b/[a+b])*100
A good example, you've got Gorzax the Destroyer vs Timmy the Eel. Timmy the Eel is an underdog and still green, while Gorzax is a very seasoned veteran. When the bets are thrown in and closed, a total of, say, 5,000gp was bet on Gorzax, while Timmy the Eel only got 100gp. The grand total of the pool is 5,100 gp. From here, you can calculate the odds.
The odds of Timmy the Eel defeating Gorzax the Destroyer:
5100gp / 100gp = 51 gp per 1 gp wagered.
Minor math error: You also get your own stake back, so the odds are 51:1 or 52 for one, because you are returned 1 gp and win 51 gp. No biggie.
The problem here is that those are the betting odds, but not necessarily the winning odds, especially if you allow the PCs to try to affect the pool. With my +30 Bluff check, perhaps I can persuade 900 gp worth of betting to move to Timmy -- and another 100 gp of money that would have been placed on Gorzax to stay home.
This means the revised pool is 4000 gp on Gorzax and 1000 gp on Timmy;. The betting odds are now 4:1 instead of 50:1.
Does this mean that Timmy's chance of winning just went up?
Of course not. I can tell all the lies I like, but the fight's still the fight.
The GM still needs some way to reasonably adjudicate the combat, and similarly some way figure out how many people would have bet on Gorzax before the PC group started putting a thumb in the scales.
The odds of Gorzax the Destroyer defeating Timmy the Eel:
5100gp / 5000gp = 1.02 gp per 1 gp wagered.So for this, it's only a .02 g (2cp) profit per 1g wagered. The odds here (.2:1, though for ease of reading, most would make this a whole number, so 1:5) would become the percentage 98%.
Minor math error: the odds are of course 0.02 :1 and not 0.2. This would more normally be expressed, of course, as 1:50. And that's not quite the percentage 98%, but 50/51 or 98.04% (which rounds to 98%).
But the big issue is this. Absent the PCs, the gamblers will presumably be betting rationally. How do we, meaning the GM, decide if the opening odds should have been 5:1, 10:1, 25:1, or 50:1 in favor of Gorzax?
That's why I suggested using that as a sooper sekrit starting point, with the GM knowing the true odds. Once he knows the odds of actual winning, he can adjust the betting odds as he sees fit....
Don't forget to set minimum and maximum bets. For example, Jimmy the Greek might have a maximum bet of 1000 gold. Gorzax the Mad might only have a limit of 100 gold, due to how much money his more generous odds cost him.
It also helps if you have most of these fights written out beforehand.
A good example, you've got Gorzax the Destroyer vs Timmy the Eel. Timmy the Eel is an underdog and still green, while Gorzax is a very seasoned veteran. When the bets are thrown in and closed, a total of, say, 5,000gp was bet on Gorzax, while Timmy the Eel only got 100gp. The grand total of the pool is 5,100 gp. From here, you can calculate the odds.
The odds of Timmy the Eel defeating Gorzax the Destroyer:
5100gp / 100gp = 51 gp per 1 gp wagered.
Minor math error: You also get your own stake back, so the odds are 51:1 or 52 for one, because you are returned 1 gp and win 51 gp. No biggie.
The problem here is that those are the betting odds, but not necessarily the winning odds, especially if you allow the PCs to try to affect the pool. With my +30 Bluff check, perhaps I can persuade 900 gp worth of betting to move to Timmy -- and another 100 gp of money that would have been placed on Gorzax to stay home.
This means the revised pool is 4000 gp on Gorzax and 1000 gp on Timmy;. The betting odds are now 4:1 instead of 50:1.
Does this mean that Timmy's chance of winning just went up?
Of course not. I can tell all the lies I like, but the fight's still the fight.
The GM still needs some way to reasonably adjudicate the combat, and similarly some way figure out how many people would have bet on Gorzax before the PC group started putting a thumb in the scales.
The odds of Gorzax the Destroyer defeating Timmy the Eel:
5100gp / 5000gp = 1.02 gp per 1 gp wagered.So for this, it's only a .02 g (2cp) profit per 1g wagered. The odds here (.2:1, though for ease of reading, most would make this a whole number, so 1:5) would become the percentage 98%.
Minor math error: the odds are of course 0.02 :1 and not 0.2. This would more normally be expressed, of course, as 1:50. And that's not quite the percentage 98%, but 50/51 or 98.04% (which rounds to 98%).
But the big issue is this. Absent...
Good catch on the latter one, though my first one (50-1) is correct. From what I remember about parimutuel betting (and what I'm re-reading), the 51 in this case includes the 1 gp you put in. In total, you get back 51 gp, but it's the 1 gp that you wagered + the 50 gp profit. So the former is correct.
Admittedly, the better mathematical formula to use is ([Total_Pool-Amount_Bet_On_Single_Fighter]/Amount_Bet_On_Single_Fighter), which with Timmy the Eel ([5100-100]/100) would land you at the true amount of profit, or 50 gp. I'm just honestly used to the former formula since I first read about this on Wikipedia long ago.
As for figuring out starting odds for fighters, I see what you're getting at. I think that's a tough one, without purposefully stacking the deck to make sure a certain person wins. I'd have to think more on it honestly...
Wow, thanks, everyone, for the quick responses. Our game is tomorrow and I wasn't sure I'd be able to get anything in time.
I'll be honest, a lot of what's on here is a little complex for me, and more than what I'm looking for. The parimutuel betting doesn't work for me, because the PCs are extremely minor players in a large pond, and tracking what else is going on with thousands of other betters seems like it would be complex. And IF the PCs do decide to bet, they'll probably want to bet multiple times (there are 30 games in a day, though only about 20 of those are bet-worthy), so I need a quick, simple, easy solution.
This is what was suggested to me by someone else. Set the odds arbitrarily. Say, 3:1 for the underdog to win, and 1:3 for the champion to win. That means you're tripling your money with an underdog bet, and getting a third again your money with a champion, bet, right? Or is that quadrupling your money with the underdog bet? You get your original stake back, right? OK, it's late and I'm tired... let's say for the sake of argument you end up with 400% your original coin. Meaning that in order to break out less than even, mathematically, there needs to be a less than 25% chance that you win that bet. So the proposed system is that the fight is narrated on the fly by me, the DM, but the winner of the fight is determined by a very simple die roll that is determined by the odds themselves, under the assumption that the people making the odds Really Know What They're Doing and that The House Always Wins (betting is done in house, not through bookies in this system, for ease). So I roll a 20 or lower and the underdog wins, a 21 or higher on a d100 and the champion wins.
Is that mathematically sound? My goal is for the PCs to have a possibility to really gamble, but I can't have their money skyrocketing out of control because it would cause problems in the campaign. And realistically, the house does always win. So I need a system that allows the PCs to easily bet, that favors the house slightly, and that is easy to set up.
Wow, thanks, everyone, for the quick responses. Our game is tomorrow and I wasn't sure I'd be able to get anything in time.
I'll be honest, a lot of what's on here is a little complex for me, and more than what I'm looking for. The parimutuel betting doesn't work for me, because the PCs are extremely minor players in a large pond, and tracking what else is going on with thousands of other betters seems like it would be complex. And IF the PCs do decide to bet, they'll probably want to bet multiple times (there are 30 games in a day, though only about 20 of those are bet-worthy), so I need a quick, simple, easy solution.
This is what was suggested to me by someone else. Set the odds arbitrarily. Say, 3:1 for the underdog to win, and 1:3 for the champion to win. That means you're tripling your money with an underdog bet, and getting a third again your money with a champion, bet, right? Or is that quadrupling your money with the underdog bet? You get your original stake back, right? OK, it's late and I'm tired... let's say for the sake of argument you end up with 400% your original coin. Meaning that in order to break out less than even, mathematically, there needs to be a less than 25% chance that you win that bet. So the proposed system is that the fight is narrated on the fly by me, the DM, but the winner of the fight is determined by a very simple die roll that is determined by the odds themselves, under the assumption that the people making the odds Really Know What They're Doing and that The House Always Wins (betting is done in house, not through bookies in this system, for ease). So I roll a 20 or lower and the underdog wins, a 21 or higher on a d100 and the champion wins.
If the odds are 3:1, then that means 3 times out of 4, the favorite wins. That's "roll a 25 or lower the underdog wins." What you described are actually 4:1 odds that pay 3:1 -- that extra bit is the house edge.
Yeah, that will work if you don't mind doing the math on the fly.
If you don't want to do the math on the fly, use multiple dice. If the odds are 3:1 in favor of the red team, use 3 red dice and 1 blue dice. Throw in a single green die for the house edge. If the biggest number comes up on a blue die, the players win.
This also lets you reverse odds easily. if they'll take bets at 3:1 against red, they'll take bets at 2:3 against blue. (Same dice pool, but only the red dice win instead.)
Dot. Most of this is over my pretty little head. :D
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What, you want it in Seussian verse?
To run a fight you first must learn
who's like to win and who's to burn.
The odds you set, by careful thought
with bookies' share NOT set at naught.
Once you've thought long and hard to see
that Gorzax' chance is set at three,
while Ripper's chance: pathetic one
a lonely slim, a singleton,
Then take the dice: for Gorzax three,
for bookies one, their greed to see,
and for poor Ripper one more bone
(how sad for wife he leaves at home!)
Then throw those dice in front of all.
The largest one get's winner's call.
What? Ripper's die? Calloo Callay!
It looks a profitable day!
For every shiny bit of gold
they bet, now three more can they hold.
But careful, now -- for in the sand
two more contestants take the stand
And so as fight goes on to fight
and cheers and jeers fill up the night,
remember that it once was said,
"In the long run, we are all dead."
What does that "house" die represent in-game? Like if you have 3:1 odds "red" vs "blue", you have four players, three players bet "red", one bets "blue", the dice comes up green, how do you explain nobody winning anything?
What does that "house" die represent in-game? Like if you have 3:1 odds "red" vs "blue", you have four players, three players bet "red", one bets "blue", the dice comes up green, how do you explain nobody winning anything?
It's a mathematical simplification.
Basically, the bookies adjust the odds so that they slightly underpay no matter which way the fight goes, based in part on the amount of money bet on the fight.
Let's assume that, rationally, Gorzax has 75% chance of winning the fight, which corresponds to a 3:1 chance of winning. A "fair" bookie, without a profit motive, would accept bets at 1:3 on Gorzax and 3:1 against him. Note that these are symmetric -- (3:1::1:3).
A realistic bookie would instead accept odds at, say, 1:4 on Gorzax and 2.5:1 (5:2) against him. Note that these odds are asymmetric.
The idea is that if people also think the odds are 3:1, then there will be about 3 dollarpounds wagered on Gorzax to every one against him. So, let's say that there's 3000 buckquid on him and 1000 against. If he wins, the fair bookie will pay out 1 buckquid for each 3 bet, or 1000 in total. If he loses, the fair bookie will pay out 3000 dollarpounds on the 1000 bet. Either way, he makes nothing.
The realistic bookie will pay out about 750 on Gorzax if he wins and therefore keep 250 as profit. If Gorzax loses, he'll pay out 2500 and keep 500.
The problem is that these kind of lopsided payments are kind of hard to calculate on the fly. So to make this playably simple, I just threw an extra die in that represents the bookies' profit.
Further to above. Another way to play it would simply be to use the fair odds, but then the bookie takes 10% off the top. Bet 1000 at 3:1, win 3000, bookie shops that to 2700. So you get back your 1000 gp stake, plus 2700 more, and the bookie keeps 300 for himself.
Further to above. Another way to play it would simply be to use the fair odds, but then the bookie takes 10% off the top. Bet 1000 at 3:1, win 3000, bookie shops that to 2700. So you get back your 1000 gp stake, plus 2700 more, and the bookie keeps 300 for himself.
That makes sense.
Wow, thanks, everyone, for the quick responses. Our game is tomorrow and I wasn't sure I'd be able to get anything in time.
I'll be honest, a lot of what's on here is a little complex for me, and more than what I'm looking for. The parimutuel betting doesn't work for me, because the PCs are extremely minor players in a large pond, and tracking what else is going on with thousands of other betters seems like it would be complex. And IF the PCs do decide to bet, they'll probably want to bet multiple times (there are 30 games in a day, though only about 20 of those are bet-worthy), so I need a quick, simple, easy solution.
This is what was suggested to me by someone else. Set the odds arbitrarily. Say, 3:1 for the underdog to win, and 1:3 for the champion to win. That means you're tripling your money with an underdog bet, and getting a third again your money with a champion, bet, right? Or is that quadrupling your money with the underdog bet? You get your original stake back, right? OK, it's late and I'm tired... let's say for the sake of argument you end up with 400% your original coin. Meaning that in order to break out less than even, mathematically, there needs to be a less than 25% chance that you win that bet. So the proposed system is that the fight is narrated on the fly by me, the DM, but the winner of the fight is determined by a very simple die roll that is determined by the odds themselves, under the assumption that the people making the odds Really Know What They're Doing and that The House Always Wins (betting is done in house, not through bookies in this system, for ease). So I roll a 20 or lower and the underdog wins, a 21 or higher on a d100 and the champion wins.
If the odds are 3:1, then that means 3 times out of 4, the favorite wins. That's "roll a 25 or lower the underdog wins." What you described are actually 4:1 odds that pay 3:1 -- that extra bit is the house edge.
Yeah, that will work if you don't mind doing the math on the fly....
The dice idea is a good one I think for NPC fights. I'd definitely use it. Obviously, for player fights, you'd want to play it out and let them duke it out.
But yeah, no matter the betting, fixed odds or parimutuel, the house/bookkeeper always takes a cut for themselves. Betting is actually fairly simple mathematics. If you want an easy way to read or translate 3:1 or any of its derivatives (3/1. 3 to 1), just use this formula.
Odds = (a:b) or (a/b) or (a to b)
Probability = (b/[a+b])*100
So for example, 3:1 becomes (1/[3+1])*100, or 25% chance of winning.
Hope that helps.
Orfamay Quest, that is a fantastic example of literary genius, and I think your simplification to my system is wonderful, as well. Thank you everyone for the assistance. :)
Now let's cheat my players out of some gold! >:D