The Tao of Gaming

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A Response to Jorbs, regarding Poker and the Tragedy of the Commons

Last night I saw that Jorbs (the Slay the Spire Streamer and former pro-poker player) posted a video discussing tournament hold ’em and the tragedy of the commons. While Tao isn’t a huge poker player, I’m not entirely without skills. And mixing game theory, poker, and policy design? The kind of catnip topic I haven’t seen in years!

To summarize the video:

  1. In tournament poker (unlike in cash games) not all chips are equal value (the Independent Chip Model)
  2. In the final table of a tournament (for example) this model leads to optimal play often being to wait for you and him to fight. “Going to war” (with random hands) on net costs both players expected value. (I don’t really think you — the average Tao reader, gifted with math knowledge far above average — needs that link. Nor does Jorbs. But maybe I’ll catch some traffic that does).
  3. Jorbs provides the example of a 6 handed table folded to the small blind. In this situation, the SB may just shove all in much more often than optimal (in a cash game), because the clever BB — even if he knows that he SB is bullying — is stuck. Calling with more than the “correct” set of hands (whatever that may be) is just destroying his expected value.
  4. This is a tragedy of the commons, a known problem (Jorbs uses the “picking up trash in a public park” example). (It’s kind of a two player prisoner’s dilemma, with a small blind/big blind, but because it can be repeated with multiple actors, it extends. There are some assumptions buried in there, but for a 20 minute video or a blog post, I think its fine to handwave this).
  5. But — here is the crux of Jorbs’s frustration — Pro Poker players are some of the most strategic thinkers on the earth (in their domain at least) … So, why have they not come up with a solution to this problem via some enforcement strategy?
  6. More frustratingly for Jorbs is that apparently poker players do not apparently acknowledge this problem.

I have many thoughts….

Let’s You and Him Fight

First, this problem is a typical “multi-player wargame” issue. This is the reason 3+ player (non-team) Chess doesn’t work. If A and B trade pawns, C is better off. There’s a reason that Titan is a classic: If A and B fight, C may be the big loser (because fighting has gains and losses …. the fight’s winner can gain points, a recruit, an angel, possibly legion tokens).

In fact, I think Jorb’s simplified model over-stated how negative the EV was of going to war. He just assumed payouts of 6,5,4,3,2,1. But typical tournament payout would be something like 300,150,75,40,25,10 … the values would depend on entries, but winning is ~40% …. Running a full EV calculation is harder then, but my gut feeling is that it lessens the impact of going all, but it would still be a negative EV play. (The calculation is harder b/c the person who doubles up now has a 40% chance of winning the tournament, but also improved chances of 2nd, and reduced chances of 5th, etc. This calculation may be solved, but I don’t know the solution and don’t care to do it now. Perhaps one of my readers knows the answer).

As any bridge player has heard, Matchpoints isn’t real bridge. (See my review of Matchpoints by Kit Woolsey). So, Jorbs feeling that tournament poker has these annoying corner cases makes total sense. You have taken an open-ended cash game (like Bridge, originally) and turned it into a format that can take an arbitrary number of entries and produce a winner in a relatively fixed time frame. Why would you expect that to be a perfect translation?

Perhaps the TL;DR of this essay is “Given that this is a known problem in tons of domains, why would you think Poker is immune?” Again — hardly satisfying. So let’s dive into it in more detail

Enforcers != Enforcement

Jorbs brings up the idea that people should enforce it. Let’s define that. An optimal player in the Big Blind will know the range of hands to calldown with if the Small Blind is playing optimally (even if I don’t). This is likely solved. If the small blind is “stealing” (betting or shoving all in on more hands than is optimal) then the Big Blind can call more aggressively and still be playing optimally (assuming he has a good estimate of how much the small blind is cheating).

We’re still in game theory. But what if the Big Blind decides to change from “optimal” play to an enforcer? Now they will not only call when it is optimal, they will call sometimes when it is sub-optimal, just to hope to catch the small blind and punish them. Something I have said professionally (but not on this blog, apparently) is

“Security is paying a small cost to impose a large cost on your adversary.”

(Me)

So, an enforcer expands their range of plays (possibly to the point of just always calling anyone who appears to be consistently stealing). If all the seats agree to do this, then you have solved the tragedy of the commons, or so the argument goes. Because players see that you are willing to punish defectors

Let’s posit that some players at the final table are just lucky and not up to game theory.

If we go back to our park example. An optimal person will pick up some trash and keep the park clean for everyone (“cooperate” in the prisoner’s dilemma). If they see a defector (someone who walks past a piece of trash without picking it up or even worse tosses some trash on the ground), they will not do anything. But an enforcer will punish the defector. Call them out, shame them, fine them, something. The enforcer takes an additional cost to make things right.

So now our enforcer rushes over to the guy who tossed a soda cup on the ground, harangues them, and then gets their reputation destroyed on social media, gets fired from their job and the litter bug’s Go Fund Me explodes … (Now might be a good time to mention the “Central Park Karen” — I haven’t followed that particular story enough to know who is actually the bad guy here, but this is not a hypothetical).

The obviously true fact is at the poker table, there’s a lot of variance. “Punishing” the defector is probably taking away a couple percent from them in the long run, but in the short run you’ll double them up a fair amount of the time.

And what do the other players see? A way to tilt the enforcer (should they ever be sitting to his right). Because “Enforcer” is another way of saying “Not playing optimally.”

Enforcement may incentivize the behavior you are trying to stop. Particularly for an opponent who recognizes he’s outclassed. (This is another aspect of Matchpoints. When you are inferior to the field you should absolutely not use the exact same bidding system as the field. Why get to the average contract and let the result be decided by technical perfection when the other players are better at it? Better to flip a coin, even if you know the coin is slightly biased against you).

It would be one thing if when you tossed some trash on the ground, enforcers (cops or otherwise) magically appeared and gave you a $50 fine. But if they magically appear and give you a fine 52% of the time and give you a $50 gift card 48% of the time, you are “losing” EV, but it might take a while to catch on.

People play lotteries voluntarily and plenty of criminals risk decades (or life) of jail time because enforcement is haphazard at best, and that’s with paid enforcers.

A minor but related point — If your village has 100 people, the park is probably small but nice. Everyone knows everyone, and if Giselle doesn’t pick up the trash because she thinks its beneath her, people will talk. If your commons are Manhattan’s Central Park …. well, there’s a lot of anonymity in the big city (except for Karen) anyway. Even if you discount the bad incentives and knew that everyone would see what you are doing and react accordingly, it matters if you are playing against the same crowd over and over again (where they will learn you are enforcing) versus some people you’ll likely never see again. (Yes, this might very well be the definition of Tragedy of the Commons, but I wanted to make it explicitly).

The Elephant in the Card Room

One aspect that Jorbs touches on …. there is an enforcer. The Casino. As he mentions, there are rules against collusion. The Casino cares about that, because if word got out that a gang (etc) were colluding in their card room the game (and their sweet, sweet rakes) would dry up. The tournament rules (like raising stakes) also exist for the Casino’s benefit, because they don’t rake each hand (only the fees), so they have incentives to make it fast enough to be profitable, but long enough that players want to play.

And the Casino is a notoriously ruthless enforcer. If I became desperate enough to resort to stealing, I’d go for a waitresses tips before trying to steal chips from an area where all the players had went to the bathroom. Even for non-crime, rules of the game enforcements, casinos are tough to beat. I’ve been called out for string raising because I didn’t know the exact rules of that particular card room (for example), even though I’m usually careful to not string-raise. The dealer is often very sympathetic to me, while rigidly enforcing the rules.

A story I read in a poker book. In one of the early tournaments, a small stack pushed all in under-the-gun. The next person (a medium stack, with several big stacks behind him) pushed all in and flipped over his pair of aces. The logic was impeccable, he was likely to bust out the small stack, but a big stack might think he was also cheating and try to bust him out, and even if he had a hand, ICM theory said it might be right to call. The ace-holder might very well grab a bunch of chips, but it was at a risk and by advertising he was making the safe solid play.

This is another weird Matchpoint-esque situation. Playing for cash you’d be happy to have a bunch of callers.

Now casinos ban players from showing their cards.

Of course, there’s a whole bunch of reasons why Casinos don’t say “No cheating in the final table, tragedy of the commons situation.” How to tell and enforce? Casinos want a bright line rule. But from the Casino’s POV, this is a legitimate angle/shot, all part of the game, and not something that (most) players care about. If players did care, Casinos might try to enforce it, but mostly it takes care of itself. Chip stacks are rarely even and the blinds will increase fast enough that other issues come to the fore.

Part of my wonders if one reason that Pros (in general) don’t care isn’t a lack of awareness, its just that its a small minor corner cases that they get over (“Matchpoints”). And against that small benefit, if they ever decided to band and somehow not have it backfire, they are worried about the casino.

Because if you (and the rest of the Pros) stood up and loudly proclaimed “We will punish defectors” some Average Joe somewhere is going to go to the Casino and say “Aren’t they colluding?” (I don’t really think this is an issue, but its an interesting angle).

Some random other thoughts / Conclusion

I keep thinking back to Hirschman’s Exit, Voice and Loyalty. For any problem inside an organization or system some people accept it (Loyalty), some complain but try to work within the system (Voice) and some just give up and walk away (Exit). This is obviously a butchering and gross simplification of Hirschman’s book, which I doubt I remember enough to treat well …. Thankfully poker isn’t nearly as important as most tragedies of the commons.

My main response to Jorbs is that I think he’s correct, and I can see why it bother him, but … well, I play a lot of Matchpoints these days. What he’s describing is true, and has no solution that I can see. If it really bothered me, well, I’d be an exit-guy as well. I’m sympathetic. Whaddya going to do? It’s the rules of the game.

PS — For a great article discussing capitalism, evolution, and various tragedies including the prisoner’s dilemma, paperclip maximizers and the race to the bottom — with a stop in Las Vegas — I suggest Scott Alexander’s (very long) essay Meditations on Moloch.

I will now jump from boring game theory stuff to what might be the closest thing to a mystical experience I’ve ever had.

Like all good mystical experiences, it happened in Vegas. I was standing on top of one of their many tall buildings, looking down at the city below, all lit up in the dark. If you’ve never been to Vegas, it is really impressive. Skyscrapers and lights in every variety strange and beautiful all clustered together. And I had two thoughts, crystal clear:

It is glorious that we can create something like this.

It is shameful that we did.

Scott Alexander, Meditations on Moloch

Written by taogaming

August 14, 2021 at 10:07 am

Among Us and my Theory on semi-cooperative games

After playing another few hours of Among Us, I remembered that I’d written a theory of semi-cooperative games (12 years ago!) and I wondered how well my theory matched up with Among Us.

Here are the original ‘rules’:

The “Cooperate/Compete” decision should be a spectrum, not just binary.

Parts of the “good” group can win without the full group.

Players must have strong incentives to act differently. These incentives should not be obvious to other players.

Clearly, among us get the last two rules right. Point two is trivial in werewolf games, which start with a good guy death, and point three is handled by the task list. The first point is (I now realize) maddeningly vague, but I think covered because Among Us is a video game and your decisions/strategies are pretty wide ranging.

And here are my ‘suggestions’ for further semi-coops:

Players should not be able to make instant decisions about each other’s play.

However, with the expenditure of resources players should be able to discover past plays. (“Tracking down evidence.”)

Once teams have been ‘proven’, the game resolves quickly.

Simultaneous play and fast turns….

Limited communication during the early part of the game….

Looking good. Obviously the differentiator in Among Us (versus the boardgames I was discussing) is the computer moderator. Play is all simultaneous, the limited resource is time/attention. In theory everyone could all stay together, do their tasks and wins. But (apart form being boring) the game makes that difficult (although not impossible):

  1. Imposters have better eyesight and can see more of the screen at a time. (Vents gives the imposters an improved speed of motion, at the cost of exposing his status if viewed).
  2. Lights sabotage drastically reduces the crewmates eyesight further (sabotaging lights)
  3. Doors sabotage splits the team directly
  4. Other sabotage forces teams to split up — O2, Reactors, etc require at least two groups to resolve.

Even with all of these, Among Us is still fragile. We didn’t have a full group of 10 (with two imposters). Two imposters makes information much fuzzier. If a body was discovered and you were with X and Y the whole time, they are clear if there’s only one imposter (and can vouch for you!). With two, they may have just been waiting for a partner to make a kill.

With 6-8 players and merely a single imposter you have to carefully tinker with settings, particularly the kill cool-down time (unlike in base werewolf a kill does not automatically trigger a meeting …. only the discovery of a body does). Too low then an imposter may able to pick off people relatively quickly. Too long and the crew-mates will often get to a position where 3 people have enough information to clear themselves, at which point the game is a lock.

And of course the settings need to be more ‘imposter friendly’ as the players count goes up. I also think that we’ll need to turn of the ‘visible tasks’ setting, which ‘proves’ that a player is a normal crew-mate with the expenditure of resources (Other people have to watch). For a single imposter, that verification is too much. In a good group, that may be too much information even with two imposters.

Overall I think that my suggestions/rules are nicely followed by Among Us — this isn’t saying that my thoughts were particularly deep, anyone who designs a good semi-coop will converge on this — but its nice to be right.

I wonder if there’s a non-obvious suggestion that would take the game to the next level, but off hand I don’t see it….

Written by taogaming

October 13, 2020 at 8:55 pm

Posted in Artificial Opponents, Game Theory

Tagged with

Screwtape discusses voting

My Dearest Wormwood,

After receiving your most recent letter, on your advice I watched the video on quick and easy voting for normal people. I am surprised that this comes as a revelation to you, since We who are down below routinely allow our charges to vote for a wide variety of things using what our patients semi-jokingly refer to as the Chicago Method (“Vote early and often”) and what your video refers to as Approval Voting.

And, as befitting our station, we scrupulously respect their votes whenever suits our mood. Which is more often than not, because all voting methods have flaws. Surely Our Father has taught you all the details of Arrow’s Impossibility Theorem, which has dozens of applications to suffering and gaming. I myself learned it at an early age.

(A more pedantic member of our kind – although I doubt you will ever encounter one – may state that Arrow’s formal proof  does not  strictly apply here. Math is a realm of The Enemy – and as such I have no done no more than dabble, lest I be accused of heresy again –  but I believe the idea generalizes. I will check with several experts I am dining on tonight).

Whenever a vote is proposed, you should of course make sure the outcome is as you desire. The stakes are high!

The video numbers make for a poor example for more interesting applications, so let us juggle them a bit. Surely even a youngster such as yourself is familiar with creative accounting?

  • The five vegetarians prefer: Veggies, Burgers (w/Veggie option), Steak (in that order)
  • The three carnivores prefer: Steak, Burger, Veggie
  • The lone Burger guy prefers: Burger, Steak, Veggie

In all cases the 1st two are “acceptable,” so burgers get nine votes, and is an acceptable compromise.

First of all, note the obvious flaw with the system. It punishes excellence. This means that, despite all of its problems, you should suggest Approval Voting whenever possible. Your goal should be to promote mediocrity and lazy thinking in all aspects. Do this consistently and your patients will always dine out on the most milquetoast and bland meals possible, never taking chances, never risking sublime beauty!

Do not mistake my critique of this system – which is done as a general exercise to instruct my favorite nephew – for a serious criticism!

Now, let us make a small change.

If, on the final restaurant named, people don’t vote on something acceptable because they prefer the currently winning option. Now, so long as Burgers are listed last, Veggies will win, because the Vegetarians, being more delighted with the currently winning option (named first or second), decline to raise their hands for Burgers. Which will now lose 5-4, despite being a unanimous winner before!

Then simply force those shuffling carnivores towards their tofu. Demand their happiness while they respect the group’s decision. Be sure to smile broadly as you choke down your okra. Sing praises towards democracy, which levels all of our patients in the same way that water always strives for the lowest resting place.

(As to my prior criticism, I simply state that while Vegetarian restaurants can be excellent in theory, much like excellent non-alcoholic beer it does not occur in practice).

As always, he who sets the vote order (and he who votes slowest, deciding after others who have raised their hands) has an immense amount of control, particularly if they well judge the preferences of others.

These tricks (along with a few more which I dare not reveal, lest this letter is intercepted) will let you control the outcome with ease, which is why we are serving a slightly maggoty meatloaf for the thousandth night in a row instead of the exquisite venison or lovely pouched trout, both clearly visible in the cafeteria.

Your affectionate uncle,

Screwtape


[H/T to Chris Farrell’s twitter feed]

My first (semi-joking) comment was that the Tao of Gaming method was to have everyone list all their options, then reject them all and walk away. This prevents mediocre games, although I admit that also has problems. I had thought I tweeted a joke about that but, much like Screwtape, I prefer the old method and send my messages encoded in the pitches and volumes of screams, although I do keep up with the times and try to limit my conversation to at most 140 screams.

An amusing coincidence — I was already thinking about the Impossibility Theorem earlier today, since my side project incorporates a quote by Kenneth Arrow in the next chapter.

Written by taogaming

June 28, 2015 at 2:59 am

Olympic Losing to Win

Slate has an article on the Badminton thing at the Olympics. (Short Form — Several teams deliberately lost matches and were disqualified by the World Badminton Federation).

The article proposes  the “Let the best winner pick their opponent” method, but any bridge player who follows the world championships knows that isn’t a Panacea. You won’t tank to go from 1st to 2nd, but you’ll tank from 4th to 5th.

The bridge system advances 8, with the 1st seed getting the pick of 5th-8th. Suppose that the teams are rated on strength (higher number is better). In theory, that means that you should have:

  1. 100
  2.  95
  3.  90
  4.   85
  5.   80
  6.   76
  7.   73
  8.   70

after the qualifying. But bridge has some element of luck. (Presumably, so does Badminton). Suppose that we say that the #3 team did poorly (but well enough to qualify) and the #7 team did great.

  1. 100
  2.  95
  3.  73
  4.   85
  5.   80
  6.   90
  7.   76
  8.   70

Now, assuming rational selection (you always want to pick the weaker opponent), The favorites pick the #8 seed (as before) but now the #2 picks the 7th seed and the #3 picks the 5th seed. Poor #4 seed (skill=85) is
playing up!. So they tank and lose a match, now look at it

  1. 100
  2.  95
  3.  73
  4.   80
  5.   85
  6.   90
  7.   76
  8.   70

Now (because they lost a match) they get to play the lucky #3 (skill 73), and the new #4 seed gets to play the #90. None of which isn’t to say that the Badminton idea is terrible, just that this isn’t as easy to solve as you’d expect.

Letting the #1 seed pick any opponent (not just 5-8) does solve this a bit, but brings up some hard feelings when the #2 seed picks the #3 seed, and the 3rd seed doesn’t get to pick.

Written by taogaming

August 3, 2012 at 6:48 pm

Posted in Game Theory, Misc

President’s Day Web Walking

  • Michael Chwe’s manuscript on “Folk Game Theory,” which analyzes Jane Austen, Oklahoma!, Brer Rabbit, and other popular stories from a game theoretic perspective, is certainly interesting. Although I’m stunned that the opening sentence of his manuscript isn’t “It is a truth universally acknowledged, that a single man in possession of a good fortune, is not in a Nash Equilibrium.”  (or some variant). [Hattip to Cheeptalk]
  • By my calculation, the odds of one partnership in bridge having NO high card points between them happens roughly once in every 21,700 hands. So that’s why it was the talk of the bridge club last week. Update — You know, there are 16 honor cards, not 12. So by my corrected calculation it’s once every 1,950,000 hands. Update Squared — Half that. Once every 975,000 hands. See below
  • I do enjoy strategy articles based off of massive data analysis of online games. Even for a trifling like 7 Wonders.
  • My pointless quest up the total fan count leaderboard for Rock Band 3 continues apace. I was going to pre-order the Squier pro-guitar from Amazon, but apparently Best Buy has an exclusive. What are ya’ll doing for that? (If anything).

So, how’s your weekend going?

Written by taogaming

February 20, 2011 at 10:42 am

The Predictioneer’s Game

I recently picked up The Predictioneer’s Game. This is yet another of the countless books trying to cash in on the Freakonomics/Malcolm Gladwell/TED trend[1]; but what the hell, it deals with game theory.

Cutting to the chase — this is more of a book to get from the library. I may go digging into some of the referenced books/articles, because I would like to see more details on modeling real world negotiations based on various desired outcomes works, but this book doesn’t have the details. (At least not yet). It is a pretty breezy read, though. Still, if you are interested in game theory and want to one-up that guy who tries to act interested in science and stuff [2], then you may be interested.

Summary — If you build a complicated game theoretical model that takes numerous actors, and rates them on knowledge, desired outcome, and how much influence they exert, you can model it pretty damn well.

(A quick glance at scholar.google.com does make me think that the author actually knows stuff. So I should be humble.)

[1] By which I mean, books that flatter you and pretend to make you think  you have some stunning insight without actually providing any details that would give you a stunning insight.

[2] Again, without actually knowing math.

Written by taogaming

January 22, 2011 at 9:39 pm

Posted in Game Theory

Game Theorist — “Dumb is the new Smart”

An interesting article points out that even game theoreticians don’t believe their own results (when money is on the line).

a research team repeated the experiment using professional game theorists playing for real money. But even among game theorists, game theory failed

One hypothesis is that you can get good results by playing dumb. If your opponent knows you are totally rational, then they have to give up a lot to keep from getting screwed. (This particular example deals with the Traveler’s Dilemma, but it applies to the Prisoner’s Dilemma, as well).

I remember a book that dealt with various puzzle aspects of Game Theory as told by Sherlock Holmes, et al. One passage discussed the prisoner’s dilemma, after a clever person tries to use it with real prisoners. When it doesn’t work, he goes to Holmes, who then sighs and calls forth one of the prisoners.

“So you know that it’s always better [to defect].”
“Yes, guv’ner.”
“Then pray explain to [this doofus] why you don’t.”
“Me mates would beat me senseless.”

Glad to see the theoreticians catching up.

Now to just figure out how this relates to unconvincing cylons, and the applications will be endless!

Written by taogaming

December 9, 2008 at 5:38 pm

Posted in Game Theory

Tagged with

Game Theorist — “Dumb is the new Smart”

An interesting article points out that even game theoreticians don’t believe their own results (when money is on the line).

a research team repeated the experiment using professional game theorists playing for real money. But even among game theorists, game theory failed

One hypothesis is that you can get good results by playing dumb. If your opponent knows you are totally rational, then they have to give up a lot to keep from getting screwed. (This particular example deals with the Traveler’s Dilemma, but it applies to the Prisoner’s Dilemma, as well).

I remember a book that dealt with various puzzle aspects of Game Theory as told by Sherlock Holmes, et al. One passage discussed the prisoner’s dilemma, after a clever person tries to use it with real prisoners. When it doesn’t work, he goes to Holmes, who then sighs and calls forth one of the prisoners.

“So you know that it’s always better [to defect].”
“Yes, guv’ner.”
“Then pray explain to [this doofus] why you don’t.”
“Me mates would beat me senseless.”

Glad to see the theoreticians catching up.

Now to just figure out how this relates to unconvincing cylons, and the applications will be endless!

Written by taogaming

December 9, 2008 at 5:38 pm

Posted in Game Theory

Tagged with

Rational Agents should Win

I don’t know if I’ve linked to Overcoming Bias before or not. It doesn’t deal with gaming, but is interesting and sometimes deals with tangential items of interest (especially to game theory).

Today they are talking about Newcomb’s Paradox (which I first encountered in a math class in middle-school. Thanks, Martin.)

And the following jumped out:

Nonetheless, I would like to present some of my motivations on Newcomb’s Problem – the reasons I felt impelled to seek a new theory – because they illustrate my source-attitudes toward rationality. Even if I can’t present the theory that these motivations motivate…

First, foremost, fundamentally, above all else:

Rational agents should WIN.

Don’t mistake me, and think that I’m talking about the Hollywood Rationality stereotype that rationalists should be selfish or shortsighted. If your utility function has a term in it for others, then win their happiness. If your utility function has a term in it for a million years hence, then win the eon.

But at any rate, WIN. Don’t lose reasonably, WIN.

{I’m adding them to the blogroll and cleaning up some old URLs).

Written by taogaming

February 1, 2008 at 5:55 pm

Basic polynomino theory?

Little Princess Tao wanted to play Ubongo. So we played. (She finished most puzzles in time, and often beat me).

This got me to thinking about polyominoes. I can look at a basic grid arrangement and a set of -ominoes and tell if it’s impossible by counting squares, and some arrangements because of parity issues. But I suspect that with some thought I could knock out more possibilities. Are there other tricks? Is there a good reference for the theory behind this that doesn’t involve massive math?

The fact that Wikipedia had nothing leads me to believe I’m spelling this wrong, or missing a technical term.

Written by taogaming

December 3, 2006 at 1:04 pm

Posted in Game Theory

Tagged with ,