Pekka
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I thought of a simple game the other day, and I can't remember if I got the idea from somewhere. It's so simple that someone else has probably invented a similar one already, but I'd like tell you about it nonetheless. At least a quick Googling for "bidding game" wasn't helpful, but that doesn't tell me much.

I propose calling it the Point Bidding Game, unless the name is already taken. I just made that name up on the spot.

It could potentially be an interesting game in the same way that Rock-Paper-Scissors can be interesting. One round of the game is up to chance, but if you play several in row, you can start forming a strategy and anticipating your opponent's moves. The game I will shortly tell you about is a little more complex as it has more choices, and you probably could use a pen and a few slips of paper when playing.

Right, so the rules I propose are short and there are two variants, which I will describe later.

It's a two-player game. You and your opponent have K rounds and N points to spend. Every round both players bid at the same time from 0 to however many points they want, just as long as their total bids don't exceed N. The player with the higher bid wins that round, and the game goes to the player with the higher total of won rounds after all K are played. Draw rounds are possible if both players bid the same number of points and hence the result can be a draw game even if you specify an odd number of rounds. (I hope the game turns out to have not too many draws.)

For example, we could pick the game's length to be five rounds and give both players 100 points. Given these numbers, possible bid sequences could be as follows.

Player A: 30 30 30 5 5
Player B: 20 20 20 20 20

This would result in A winning three to two. You should also note that when there are five rounds, there are really only four rounds where you can make a meaningful choice about what to bid, since it doesn't make sense to not bid all the remaining points on your last round.

The two options I mentioned are that you could pick each new bid after the last round, or you could pick all the bids ahead of time in one go, and then just see who wins most rounds. I think the first option would make for a more exciting game, since you can make choices after seeing what your opponent does.

Now, I am not at all sure if this game ends having a lot of draws if you play well, or it turns out to be boring for other reasons. I don't even know how to play it well yet. So, I wanted to present it here and see if anyone wants to analyze it further or comment otherwise.

Now, my three questions are:

Who has invented or played a game like this before me? Surely such a simple idea has occurred to others too.

What are good strategies for either variant of the game?

Which rules option makes for a more exciting game?

Also, I'm sure people can think of improvements to this game. If you do, tell us what rules you'd like to suggest and why they are good ideas.


(I was thinking that instead of many points, you'd have a few dice to bid and then you'd have to roll and sum them each round to see who actually won. More randomness could make the game more exciting. Or it could make it less exciting. But that's just speculation.)

Red-XIII
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Mhh I don't think I ever heard this but that sounds interesting.
Anyway there's something I didn't understand, for exemple there are 5 rounds: we have to show our points choice every round or all at once at the end? IMO the former will be more intriguing and the latter would be more intuitive. I like these both these choice, which choice you thought while thinking about this game?

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[Last edited by Red-XIII at 11-12-2012 08:04 PM]

Pekka
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Red-XIII wrote:
I like these both these choice, which choice you tohught while thinking about this game?

My original idea was taking each turn one at a time. That is, you'd both write the bid on a piece of paper and reveal them at the same time during each turn. But it wasn't much of a leap to think that maybe you could write them up all at once. The latter way would allow for playing more games in the same amount of time so there'd be some advantage to it.

I suppose you can try to play both ways and find out which one works best for you

Red-XIII
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I see, I'll definitely try this game and I'll let you know if I find some interesting rules

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Bombadil
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Seems interesting. I'll try to play it tomorrow at lunch with my coworkers...

It reminds me of For Sale, a card game. In For Sale, each player starts with a fixed amount of money. In the first round they bid for different properties (of different value) similarly to what you propose. There is a second round where they try to sell what the have bought in a similar manner.

You can find video/written reviews of the rules at boardgamegeek, and probably on the forums you can find similar games to for sale or your game:
http://boardgamegeek.com/boardgame/172/for-sale

stigant
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Interesting...

Here are my initial thoughts:
0. Place all your bets ahead of time seems uninteresting to me since it's really all chance, so I'll only be thinking about the one bet at a time version unless someone convinces me that I'm wrong about all-bets-at-once version.

1. There is no winning strategy at the beginning of the game (assuming that both players start with the same amount of money) since the game is completely symmetrical.

2. Looking at small numbers of turns (left):
a. If the players have the same amount of money going into the 2nd to last turn, then the last two turns will be a push (either each player will win one turn or they will tie on both turns).
b. If one player has more money going into the 2nd to last turn, he's guaranteed to win at least one of the two turns, so his goal should be to win both. On the other hand, the other player is guaranteed to lose one turn, so he should play to win one of the turns. I'm looking to see if I can find a optimal strategies in this case.
c. If there's a good way for the player with more money to win the last two turns, it seems likely that good strategy (not winning all the time, of course) would be to tank the first (K-1)/2 turns, and then win the last (K+1)/2 turns.

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[Last edited by stigant at 11-14-2012 03:01 PM]

Someone Else
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Well, the person with less would want to bet it all on one of them, forcing the other player to guess which one he's going to bet on.

Pekka
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stigant wrote:

Here are my initial thoughts:
0. Place all your bets ahead of time seems uninteresting to me since it's really all chance, so I'll only be thinking about the one bet at a time version unless someone convinces me that I'm wrong about all-bets-at-once version.

OK, I'd like to clarify things on this issue. I agree that the all-bets-at-once is uninteresting when played only once, but if you play multiple consecutive games, you can possibly pick up on the patterns your opponent uses and outguess him or her, thus winning more games overall.

You could then argue that just playing randomly would be hard to beat, and it is possible that this is true. (I haven't worked it out, though. Open question: Is random bidding hard or impossible to beat in thsi variant?)

However, when combined with a round robin tournament system and multiple participants, I would expect that the top players would not be the ones playing randomly, but the ones who could gain the most from beating the weaker players with larger margins than the random players. This assumes the tournament used some sort of tie-breaker based on the results of the individual series of games.

But that's mainly speculation. I think we should focus on the single sequential bid version for now, and consider placing multiple bids at once a possible variation. You could also combine the two ideas, so you'd place K multiple bids each round during L sequential rounds for a total of K * L bids, but let's not make the game that complicated yet.

So, I think we can assume that in this thread we'll talk about the version where you each place one bid at a time and know the results of the earlier rounds when placing later bids, unless specified otherwise. I did want to mention I was also thinking if this game would work as a tournament game, too, which is actually a separate consideration from whether a single game between two players is interesting.

It's of course perfectly valid to consider just two poeple playing a few games or a single game too. So this is not a complaint by any means.

I have read all the replies, but I don't have enough time for more comments yet.

stigant
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OK, I'd like to clarify things on this issue. I agree that the all-bets-at-once is uninteresting when played only once, but if you play multiple consecutive games, you can possibly pick up on the patterns your opponent uses and outguess him or her, thus winning more games overall.

That's certainly possible, but I'd argue that it's still uninteresting (at least from the standpoint of finding optimal strategies) since the game is now psychological and thus won't really bend to mathematical analysis. The best players will change their strategy between games (knowing that their opponents will analyze their playing patterns and adjust accordingly), and it's silly to try to find optimal strategies for beating sub-optimal players.

Well, the person with less would want to bet it all on one of them, forcing the other player to guess which one he's going to bet on.

So if that's the best strategy for Poor, then Rich's best strategy is to bet Poor's stack + a little on one round (chosen randomly). That gives Poor's strategy a 50% chances of "succeeding" (ie winning one round). Can he do better?

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TripleM
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Well, there's several issues at play here.

There's going to be psychology involved regardless, since if the poorer player announces what he will bid, the richer player can always win by bidding a tiny bit more. And even if the poorer player chooses a probabilistic strategy, then there will be an optimal strategy for the richer player; but then knowing that the poorer player may have a better strategy to combat that, and so on.

Plus of course, since ties are possible you have to weigh up whether certain probabilities of winning/tieing/losing are better than others.

Suppose we take stigant's example, with two rounds, one player has k>1 times the other player, the weaker player wins if they win 1 round, and the weaker player announces a strategy first.

Then with optimal play the richer player will win with probability k-1 (or always, if k>=2, of course).

This is because if the weaker player chooses to bid r in the first round and 1-r in the second round, the second player wins with (x,k-x) if x-(k-1) < r < x. In other words, he can basically choose any range of size k-1 - the best the first player can do is ensure all ranges are equal in total.

Conversely, if the richer player announces a strategy, we get the same result. But if one player announces a strategy, the other player believes them and places the best bid, then the initial player can do better by changing their mind.

So who knows what I'd actually do if I were playing, even in this simple case.

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[Last edited by TripleM at 11-16-2012 04:34 AM]

Pekka
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Here is a draft of the game rules. Please suggest improvements, point out infelicities and make any use of them as you will. I want to grant this rules text and the game to the public domain if it turns out it is an original invention, and if it is not, I hope to hear who else has studied a game like this before me. If you pass it on, you can credit me in a way you wish (even not at all).

I haven't tried playing or analyzing this seriously yet, but I think I can fairly confidently say that playing randomly (meaning that you select your bids randomly so that any point is equally likely to be bid in any possible round) doesn't appear to be a good strategy if your opponent figures it out. She can then simply throw a few rounds by playing zero and quite probably win the rest, which conveniently should amount to just over the half of the rounds.

I also hope to hear from anyone who managed to get people to playtest this seriously.

Sequential bidding game rules

Equipment: The recommended equipment is a sheet of paper and a pen for each player.

Number of players: Two or more. These rules describe how a single two-player game is played. You can set up a tournament with multiple games if there are more players.

Parameters: For each single game, both players receive a fixed amount of points and there is a set number of rounds. (For a quick game, I suggest using 30 pts and 3 rounds, and for a medium game 100 pts and 5 rounds for each player. More playtesting might suggest different figures.)

Playing the game : A single game consists of N rounds that are played sequentially. Before each round both players decide on how many points they want to bid. The bids are then revealed simultaneously and the higher bid wins the round, or if both players bid the same number, the round is a tie. This is repeated N times with both players always being aware of how many points the opponent bid during the previous rounds.

The allowed number of points to bid is from 0 to the total points left for the player. After each round the last bid by each player is subtracted from his or her points. All the bids from each round of a single game must add up to the total agreed before playing. (Note: This effectively means you must always bid all your remaining points during the last round, if you have any left.)

It is suggested that when playing a single game, you write your bids in a column marked with the round numbers, as each bid is made. After playing the game both players can easily verify that the bids were legal by adding up the numbers.

Scoring: Each round won by a player increased his or her score by one. If a tie occurs, both players get half a point. The player with the higher total points from all the rounds of a game wins the game. If both players receive the same score, the game is a tie.

Variants

Variant 1: Multiple bidding targets

Instead of making a single bid each round, the players have multiple targets named with letters. For example, there could be three targets named A, B and C during each round. The players split their bid total during a round into these targets and each target is compared to its equivalent opponent's bid. As previously, each higher bid earns the bidder one point and a tie earns half a point for both. The higher total of points wins.

Variant 2: Bidding dice

To add randomness to the game, give each player a set number of dice to bid for each game. During each round the player rolls the number of dice he or she bid and and the effective bid for that round is the total from this roll. The results are calculated from the effective bids the same as in the standard game. (I suggest using only a few six-sided dice for each player to avoid a lot of rolling and calculation.)

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[Last edited by Pekka at 11-19-2012 12:43 PM]

Znirk
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Pekka wrote:
Who has invented or played a game like this before me? Surely such a simple idea has occurred to others too.

There's a similar mechanism in the boardgame Adel verpflichtet: players compete for collectible antiques and, in one part of the game, try to outbid each other in auctions by means of hidden, simultaneously-revealed bids. There are significant differences though: on every turn there's a choice of whether to even compete in the auction (or do something else), and the subdivision of the available total "money" into biddable chunks is given by the game equipment (different preprinted "cheques"), not freely chosen by the player.

mrimer
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That name seems familiar. Is that Hoity Toity in English?

edit: yes, seems so

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[Last edited by mrimer at 11-19-2012 10:47 PM]

Tahnan
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Pekka wrote:
I haven't tried playing or analyzing this seriously yet, but I think I can fairly confidently say that playing randomly (meaning that you select your bids randomly so that any point is equally likely to be bid in any possible round) doesn't appear to be a good strategy if your opponent figures it out. She can then simply throw a few rounds by playing zero and quite probably win the rest, which conveniently should amount to just over the half of the rounds.

I wonder how true that is; if my random numbers happen to fall out such that the first several rounds, the ones my opponent is throwing with a zero bid, are rounds where I'm betting only a point or two, then by the time my opponent stops throwing rounds, I'm not sure she'll "quite probably" win the rest.

At any rate, I wanted to throw out for comparison a game that I believe comes from a book by James Fixx called "Games for the Superintelligent". The rules are as follows: take a deck of playing cards and divide it into suits. Shuffle the hearts and put them in a face-down pile in the middle of the table. Give the clubs to one player and the spades to the other. On each turn, flip over the top heart; both players then select a card from their hands to bid on that heart (aces are worth 1; jack/queen/king are 11/12/13), which they put face down in front of themselves and then flip simultaneously. Whoever plays the higher card wins the heart (the card is split on ties), and the cards played are removed from the game, but left face-up so each player can see which cards the other has played. Whoever has the higher number of hearts at the end wins.

A fairly obvious strategy would be to "bid" whatever the card is worth: bid 1 (by playing your ace) for the ace, 2 for the 2, and so forth. Of course, if your opponent knew this was your strategy, they could wipe the floor with you by bidding two for the ace, three for the two, and so on up to 13 for the queen, and then one point for the king; you win the king, and lose the game 13 to 78. In general, you want to outbid your opponent by one when you win and underbid your opponent by as much as possible when you lose--there's a wonderful smug feeling when your opponent puts out their king as a bid on something high-scoring and you put out your ace, giving them the high-scoring card but rendering your own king unbeatable for whatever you care to take later on. (Though if they see your king coming, they'll put out their ace, and won't you feel stupid...)

There are obvious differences from Pekka's proposed game: in this game (whose name, alas, I cannot remember!), while both players have the same number of points, i.e. 91, to use over the thirteen rounds, they can't distribute them freely but have to play them in specified units; scoring isn't merely based on the number of rounds won, but instead the number of points won. Still, it ought to serve as an interesting basis for comparison.

Pekka
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Tahnan wrote:
I wonder how true that is; if my random numbers happen to fall out such that the first several rounds, the ones my opponent is throwing with a zero bid, are rounds where I'm betting only a point or two, then by the time my opponent stops throwing rounds, I'm not sure she'll "quite probably" win the rest.

It will be very rare to make small or large bids with the kind of randomness I described. Another sort of random function for choosing the moves might perform better, but in this case my experiments showed that playing randomly with 30 pts and 3 rounds only resulted in the occasional bid of 15 or above for the random player, meaning the player playing two fifteens won almost every game.

For 100 pts and 5 rounds the random player fared even worse. The specifically chosen counter strategy won about 91 to 92 per cent of the games with 30/3 and approximately 99.6 per cent with 100/5.

But you could have been thinking of another way of randomizing the moves instead of picking a random round for each point to play in, which results in a rather even spread most of the time. This way is beaten "quite probably" with a specifically crafted counter strategy. That was all my simple script tested.

So, just to be clear, my claim isn't that playing randomly is always easy to beat, but that this particular random strategy is easy to beat if you can anticipate it.

At any rate, I wanted to throw out for comparison a game that I believe comes from a book by James Fixx called "Games for the Superintelligent".

Thanks for pointing that out. If anyone can remember or find the game's name that would be helpful too, but at least it's good to know a similar idea has appeared elsewhere. (I'll see if I can find out more myself, too.)

Tahnan
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File: pekka_game.py (2.3 KB)
Downloaded 39 times.
License: Public Domain

Pekka wrote:

Tahnan wrote:
I wonder how true that is; if my random numbers happen to fall out such that the first several rounds, the ones my opponent is throwing with a zero bid, are rounds where I'm betting only a point or two...

It will be very rare to make small or large bids with the kind of randomness I described.

Oh, I see! I admit I didn't read what you posted carefully enough--I assumed that "playing randomly" meant "assigning a random number of points to each round" as opposed to "assign a random round to each point". The latter feels much less random to me, in fact, insofar as--as you noted--the more points you have, the more likely they are to distribute evenly. The strategic play of [0, 15, 15] is certainly likely to beat something that's going to look roughly like [10, 10, 10]; the strategic play of [0, 0, 33, 33, 34] is even more likely to beat something that's going to look roughly like [20, 20, 20, 20, 20].

That's why I read "randomly assigning points" as picking a random number of points (out of the points left) for each round, using everything that's left for the final round. That's a case in which something like [2, 20, 8], which beats [0, 15, 15], isn't nearly as unlikely as it is in the pick-a-round-for-each-point random approach. Granted, even this random strategy isn't terribly good: on the 30-point 3-round game, the [0, 15, 15] strategy wins something like 65% of the time, and on the 100/5 game, the [0, 0, 33, 33, 34] strategy wins around 84% of the time, which does indeed amount to "quite probably winning the rest", though it's at least somewhat better than the other random strategy.

Just for the heck of it, I wondered what happened if you pitted these random strategies--let's call them the Pekkarandom strategy and the Tahnanrandom strategy--against each other. Ironically and perhaps paradoxically, Pekkarandom beats Tahnanrandom with about the same percentages that the "strategic" play does: 65% for the 30/3 game, 88% for the 100/5 game! I didn't expect that Tahnanrandom, which does notably better against the "throw-the-first-half" strategy than Pekkarandom, would lose so badly to it.

Python script attached, for anyone who wants to check or vary my work.

Pekka
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Thanks for the script Tahnan. I don't know when I can follow up on this thread properly the next time, but I will do so eventually. Meanwhile, I'm at least reading all the comments.

brian_s
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If there is enough interest, I will host a game to see how this actually plays out. I have a spreadsheet setup for a six round game with a 100 bidding point tally. If you have played in any of my Battleship games, you can expect the same style of organization.

Scoring will be a little different than Pekka proposed. My scoring rules: players' scores for each round will be the number of other players outbid. For example, five players submitting bids of 12,9,12,19,17 for a round would recieve 1,0,1,4,3 score points respectively.

There will be five submission rounds, the last one being the final split between round 5 and round 6 bids. Anyone not submitting an entry in a timely fashion will have a bid of 0 placed for that round. Nonsubmission in the 5/6 round will result in two zeros being placed.

I'll wait until the end of the month for people to sign up. On January 1st, the game will begin if enough people want to try.