ArticlesThe Set-Up
It’s a corporate Christmas party, around 1999. The dot com crash hasn’t happened yet, the party-organisers have a considerable budget, 200-300 guests, and an idea of a good time strongly influenced by the company’s sales and marketing teams – people who have a tendency to differ from my idea of a good time.
After the main course and a few speeches, the MC explains the entertainment for the evening.
A number of casino tables and croupiers have been hired for the night – roulette, blackjack, craps and the like. Each of the party-goers is issued with $100 worth of fake money, which can be turned into chips and gambled.
After an hour or so of gambling at the over-crowded tables, everyone will return to their seats to use the fake money to bid in an auction on a few reasonably-valuable prizes.
Ok, thinking caps on… what is a good betting strategy here? Give it some thought before reading on.
Roulette: A strange game. The only winning move is not to play.
If you’ve been listening to your maths teacher, you will probably know that (with a few exceptions) casino games are a losing proposition. The maximising strategy is bet zero dollars.
What are the exceptions? I can think of: corrupt or incompetent croupiers, unbalanced roulette wheels, card-counting in some BlackJack variants and being the house!
Now, I have seen a croupier, hired for a 21st party, with no real money being bet, attempt to impress a cute blonde by quietly dealing her 4 aces in her first ever game of Texas Hold’Em, and brag to her about it afterwards. So I don’t doubt the existence of corrupt croupiers, but the tables were surrounded by witnesses and bribing a croupier to cheat your workmates is frowned upon by Management, so let’s eliminate that.
An hour is hardly enough time to detect if and how the roulette wheels was unbalanced. So let’s ignore that.
Card-counting on BlackJack was possibly an option, but – in the cases where the rules even allow it to be profitable – that is only moving the edge slightly in the player’s favour. Within an hour of playing – especially on a busy table with lots of novices, and hence few hands played, – that edge will be hidden in the noise.
No, it seems if you want to maximise your expected outcome of fake dollars, you are better off standing back from the tables, and watching.
But gambling is fun!
If you get a thrill of the chase from gambling, and don’t have a compulsion problem, there is an argument here to go for it, in this innocuous situation.
You aren’t risking much; your expected value from the whole competition before you start is the value of the prizes (to you) divided by the number of party-goers. I don’t remember exactly what the prizes were, but that number was probably $5-$10.
Craps is fun! It’s a good night’s entertainment.
Well, at least, it is for some people. I enjoy playing games of skill, but games of pure luck don’t interest me much. I didn’t actually want to push through people to get to the tables. For me, conversation is fun, and I was happy to talk to the people left behind.
What measure are we trying to maximise?
Well, based on the previous point, the idea is to maximise fun – which is why some people should go gamble, and some people should sit and talk, independent of the competition. But this post is about the competition, so let’s get back to that.
The idea is to maximise the number of prizes you get, not the number of fake dollars. The fake dollars are really just points. In two hours, they are worth absolutely nothing.
Your normal gambling strategies (such as Martingale, Kelly, Labouch̬re РPro tip: Two of these are mathematically bogus, and the other recommends not betting at a casino.) are designed to try to carefully win small amounts. They want you to walk away with a few more dollars than you started with.
However, in this competition there is an auction at the end. It is winner takes all (or at least, the top few winners share a bounty of a prize each; worse if the top grossing player has more than twice their next rival.)
At a regular casino, if you walk out with $110 after betting $100, you are a winner. If you walk out with $100, without betting any of it, you are a smart person, and doing better than average. But in this competition, 100 points is worthless; it isn’t likely to be in the top few players.
The goal is not to gamble to maximise your expected (in the mathematical sense) dollars, but to maximise your chances of being in the top few rankings against your competitors.
All this was going through my mind, a decade ago, as I watched those competitors scramble for positions at the tables.
Social Factors
I kicked myself later for not factoring in another key aspect – that points were transferable, and people are social animals.
As people walked away from the tables with $50 or so, when the realisation dawned on them that the fake money was now worthless to them, they gave the money to their friends with the most chance of winning a bid.
These parcels of money started to snowball, into larger parcels.
A competitor didn’t need to win – they just needed to convince a large group of gullible friends that they had won – in order to come into a sizable windfall after the gambling had finished – especially, if they could cast themselves into a underdog position, well-deserving of a little boost to take on the big winners.
The Big Unknown
How much money would it take to be competitive in the auction? Most people would lose money on the games – that’s how casinos pay their bills, after all. It was likely that some people, betting their normal strategies were likely to do well – perhaps doubling or tripling their money. Combine that with the donations, maybe they would double that again.
Maybe there were a few others out there, like me, who could see the bigger picture, and were betting with the goal of maximising their chances of being in the top few.
I made my decision.
My Betting Strategy
I elbowed my way to the roulette table, converted my fake $100 bill into chips, and placed them all on number 18.
Winning would mean I would walk away with $3600, and a very good chance of taking one of the auction items. Losing would mean I was out of the game. A 1 in 37 chance of winning a prize was much better than my original position.
With this in the back of my mind, I waited for the other bets to be laid, the ball to spin and erratically bounce until it settled into a slot.
…
Of course, it wasn’t at all surprising that my number didn’t win. This is a real story, not a Hollywood movie. Defeated, I pushed back through the crowd and returned to the conversations happening away from the tables.
But I stand by my approach.
Coda
I don’t recall how much the auction items went for. In fact, I have a suspicion I left the party before the auction was held. It’s a shame. It is the one data-point that would be good to know if this ever happened again. If the auction items went for $500, then I need not have taken such a big risk; I could have spread my chips over several numbers. If they went for more than $3,600, I would have had to go for more risk.
Coda To The Coda
This story is ostensibly about gambling, but the reason it came to mind is that I have been pondering the design of an AI (artifical intelligence) player for a simple game – totally unrelated to gambling activities.
I am aware that the AI should play looser, make riskier moves with higher payoffs, when it finds itself in a losing position. Even if the expected point value of the game move equivalent to “put it all on 18 equivalent” is less than the expected value of the game move equivalent of “keep your money in your pocket” it may be worthwhile. The true goal is not to maximise your expected points, but to maximise your chance of having more points than your opponent the end of the game.
Alas, I haven’t worked out the right approach yet to do this analysis: I need to be able to answer the question: “If I need 500 points to win, and my competitor needs 225, should I make a move that guarantees me 200 points, or one that has a one in three chance of getting me 400, and two in three of nothing.” I think this question is going to be harder to solve that the entire rest of the AI.
Comment by Sunny Kalsi on July 18, 2010
There’s a series about video game design which may be of use to you. There’s a part which talks about a game being about decision making. Instead of a strategy which tries to maximise some utility function, try and make your AI switch to making decisions (or form and alter a strategy). Your “strategy” in this case was to gain a dominant position in the money stakes to make the purchasing phase easier.
Mine would probably have been to sit back, opportunistically take advantage of social pooling, and use the auction stage to drain resources from the other players to gain the upper hand on a prize I really wanted. Of course, if you’d succeeded in your strategy mine would’ve failed against it, but who knows.
The key here is that the decision making for “how do I win this game” comes right at the beginning, and then is altered based on other people’s decisions. “I see this other player’s strategy is going to cause mine to fail, I should alter my strategy now.”
I should mention “mirror neurons”, the neurons which effectively “imagine yourself performing another player’s actions”. It has been linked to empathy, and perhaps to more things. Effectively back-tracking from decisions to the other player’s strategy could allow the AI to alter their strategy.
Comment by Julian on July 18, 2010
I spent an interesting half-hour skimming the course that you suggested. It wasn’t focussed on video games, but general game design. I couldn’t find any reference to AI bot design there.
Certainly the bot design is about decision-making. That’s pretty much its only function. And, yes, I was initially thinking that the maximising of the utility function (i.e. trying to get the biggest point score per decision) was going to be the challenge, where I now believe it is more about responding to the other player’s position. Now arguably it should also respond to the other player’s perceived strategy, but I don’t think that is required for this particular game, where there is no interaction between the players.
Your assumption with the resource draining strategy is that other people aren’t doing the same, and that the item that interested you most wasn’t up first. The final auction item price would be the most capricious – bids would be reduced by the money spent earlier, but maximised because everyone left would have no reason not to bid their entire funds. If people who hold back waiting for the last item, they may find they have inadequate funds and no chance to bid on a consolation prize.
I did try to factor the behaviour of others into my decision that a 3600% percentage win would be sufficient. If all of the players had followed a tactic similar to mine, then the bids would have been on the order of
(number of people / number of items) * 100%– say 5000%.My reasoning here is that if there are 250 people, and 5 prizes, and everyone takes 1-in-50 chances, there will be about 5 winners, who can bid. It gets more complicated than that, but good enough for a first approximation. A 1-in-50 chance should multiply your money 50-fold, approximately.
However, I reduced my risk-taking that based on my estimates of the likely sub-optimal behaviour of the other players. I wish I knew what the bids finally were, because I suspect I still took slightly too much risk. Maybe I should have bet half my money on 18 and half on 19.
Your link on mirror neurons suggests that the results have been ignored, which made me laugh. I’ve read and heard dozens of articles and podcasts that describe them and overstate their explanatory power for many different aspects of animal behaviour. I really can’t see their relevance here. The ability to have a Theory of Mind about other players is a key aspect of game-playing and AI design, but doesn’t require the existence of mirror neurons.
Comment by Julian on July 18, 2010
I note that in the post and comments on Battleship Strategy, some of these topics came up, including: changing behaviour based on the perceived playing style of the opponent, and the appropriateness (or not) of taking larger risks against certain players.
Comment by Sunny Kalsi on July 19, 2010
Sorry about that. It’s really quite a large course, and I really should’ve found the exact place where decision making is cited. However, there are a lot of interesting things mentioned there. It deals with the game design portion of a game, you’d have to look at it from the player’s end.
Yeah the draining idea is a bit flawed, but there are a lot of variables: Do I want one prize? Do I want to maximise the number of prizes? Is it worth going for an all-or-nothing approach? These all need to be taken into account. I was thinking all the prizes were roughly equivalent utility, or the best prize was left till last. You just bid low for everything, and sense for whether the other person actually wants the other prize or not. Then you press your luck.
The way other people play dramatically alters strategy, depending on the game. I remember playing Bartog with some high school friends, thinking that it really shouldn’t be played like that (but of course it’s just another way of playing it). Players must interact somehow. Even a game like Dominion is all about making an aggressive push based on the push other players make, even if your deck and moves are technically unaffected by theirs.
Yeah I raised an eyebrow to the mirror neurons go ignored thing as well. The conclusion was what I was after — the idea that you could gain an uncanny sense on your opponent’s intent by figuring out a way of having the AI “imagine” themselves playing the moves out. I was thinking maybe using something like mirror neurons might yield an efficient (or somehow superior) implementation to gain a Theory of Mind in your AI.
Comment by Richard Atkins on July 30, 2010
Some recollections about that party:
I vaguely remember the blackjack dealer’s required play style being more lenient than the standard casino ones: after all, they didn’t care if we took all their fake money.
Did they have a chocolate wheel? If they’d had one, the top square on that probably would have been the best expected value compared to any other game – after all, most casinos have a 100x prize on a 32 slot or so wheel. That’d give an even better return (on average!) than your roulette strategy.
[Note: I censored the surname of the then Regional Managing Director. I’m being overly paranoid, but I don’t particularly want to associate this blog with my former employer. – Julian]
Comment by Julian on July 31, 2010
Richard, thanks for the precision. I couldn’t remember exactly, which is why “about 1999”.
Secret prizes? Ugh. I definitely left before then, because I would have remembered that.
Pretty sure the Chocolate Wheel/Wheel of Fortunes don’t have a 100x prize on a 32 slot wheel, or that’d be my living, betting on the wheel all day. More typical is a 40x prize on a 54 slot wheel.
Comment by Alan Green on August 14, 2010
Fascinating. Our office is due to have a casino night this coming week. We’ve been told, “There will be 1 winner on the night with a Casino Champion crowned.” I suspect that, like the event of 9.5 years ago, there will be room for social hacking.
Expect tens – or even dozens – of hits as my office mates scour this article for advice. I hope you have the server capacity 😉