# On Robin Hanson’s Board Game

An Anal­y­sis Of (Robin Han­son at Over­com­ing Bias): My Mar­ket Board Game

Robin Han­son’s board game pro­posal has a lot of in­ter­est­ing things go­ing on. Some of them are re­lated to cal­ibra­tion, up­dat­ing and the price dis­cov­ery in­her­ent in pre­dic­tion mar­kets. Others are far more re­lated to the fact that this is a game. You Play to Win the Game.

### Rules Summary

Dol­lars are rep­re­sented by poker chips.

Me­dia that con­tains an un­known out­come, such as that of a mur­der mys­tery, is se­lected, and sus­pects are picked. Play­ers are given \$200 each. At any time, play­ers can ex­change \$100 for a con­tract in all pos­si­ble sus­pects (one of which will pay \$100, the rest of which will pay noth­ing).

A mar­ket is cre­ated for each sus­pect, with steps at 5, 10, 15, 20, 25, 30, 40, 50, 60 and 80 per­cent. At any time, each step in the mar­ket ei­ther con­tains dol­lars equal to its prob­a­bil­ity, or it has a con­tract good for \$100 if that sus­pect is guilty. At any time, any player can ex­change one for the other – if there’s a con­tract, they can buy it for the listed prob­a­bil­ity. If there’s chips there, you can ex­change a con­tract for the chips. Who­ever phys­i­cally makes the ex­change first wins the trade.

At the end of the game, the win­ning con­tract pays out, and the player with the most dol­lars wins the game.

### Stages of Play

We can di­vide play­ing Robin’s game into four dis­tinct stages.

In stage one, Setup, the source ma­te­rial we’ll be bet­ting on is se­lected, and the sus­pects are gen­er­ated.

In stage two, the Early Game, play­ers re­act to in­cre­men­tal in­for­ma­tion and try to im­prove their equity, while keep­ing an eye out for con­trol of var­i­ous sus­pects.

In stage three, the Late Game, play­ers com­mit to which sus­pects they can win with and lock them up, sel­l­ing off any­thing that can’t help them win.

In stage four, Re­s­olu­tion, play­ers again scram­ble to dump now-worth­less con­tracts for what­ever they can get and to buy up the last of the win­ning con­tracts. Then they see who won.

### Setup

Not all mys­ter­ies will be good source ma­te­rial. Nor do you ob­vi­ously want a ‘cer­tified good’ source. That’s be­cause know­ing the source ma­te­rial cre­ates a good game, is a huge up­date.

A proper mul­ti­ple-sus­pects who-done-it that keeps ev­ery­one in sus­pense by de­sign keeps the scales well-bal­anced, en­sur­ing that early re­s­olu­tions are fake outs. That can still make a good game, but an even more in­ter­est­ing game car­ries at least some risk that sus­pects will be defini­tively elimi­nated early, or even the case solved quickly. Com­edy rou­tines some­times re­fer to the is­sue where they ar­rest some­one on Law & Order too early in the epi­sode, so you know they didn’t do it!

When watch­ing sports, a similar dilemma arises. If you watch ‘clas­sic games’ or oth­er­wise en­sure the games will be good, then the first half or more of the game is not ex­cit­ing. Do­ing well early means the other team will catch up. So you want to choose games likely to be good, but not filter out bad games too ag­gres­sively, and learn to en­joy the oc­ca­sional de­mo­li­tion.

The setup is also a give­away, if it was se­lected by some­one with knowl­edge of the ma­te­rial. At a min­i­mum, it tells us that the list is rea­son­ably com­plete. We can cer­tainly in­tro­duce false sus­pects that should right­fully trade near zero from the start, to mix things up, and likely should do so.

One solu­tion would be to have an un­known list of con­tracts at the start, and in­tro­duce the names as you go along. This would also po­ten­tially help with pre­vent­ing a rush of trades at the very start.

In this model, you can always ex­change \$100 for a con­tract on each ex­ist­ing sus­pect, and a con­tract for ‘Some­one You Least Sus­pect!’ Then, when a new sus­pect is in­tro­duced, ev­ery­one with a ‘Some­one You Least Sus­pect!’ con­tract gets a con­tract in the new sus­pect for free for each such con­tract they hold. There are sev­eral choices for how one might in­tro­duce new sus­pects. They might un­lock at fixed times, or play­ers could be al­lowed to in­tro­duce them by buy­ing a con­tract.

The com­plex­ity cost of hid­ing the sus­pects, or let­ting them be de­ter­mined by the play­ers, seems too high for the de­fault ver­sion. It pro­tects the fun of the movie and has some nice prop­er­ties, but for the base game you al­most cer­tainly want to lay out the sus­pects at the start. This gives a lot away, but that’s also part of the game.

For the first few games played, it prob­a­bly makes sense to choose mys­ter­ies ‘known to be good’ such as a clas­sic Agatha Christie.

The game would pre­sum­ably come with a web­site that al­lowed you to in­put a movie, show or other me­dia, and out­put a list of sus­pects. It would also want to ad­vise play­ers on whether their se­lec­tion was a good choice, or sug­gest good choices based on se­lected crite­ria. Both will need to be bal­anced to avoid giv­ing too much away, as noted above; I’ll talk more about the gen­eral ver­sion of this prob­lem an­other time.

If you are in charge of setup, I would en­courage in­clud­ing at least one sus­pect that ob­vi­ously did not do it, in a way that is easy to rec­og­nize early. This pre­vents play­ers from as­sum­ing that all sus­pects will re­main in play the whole time, and re­wards those pay­ing at­ten­tion early. Keep peo­ple on their toes.

### The Early Game

The mar­ket maker is in­ten­tion­ally dumb, al­though in de­fault mode they are smart enough to know who the sus­pects are. All sus­pects start out equal.

There are a bunch of good heuris­tics, many of which should be in­tu­itive to many view­ers of mys­ter­ies, that cre­ate strong trad­ing op­por­tu­ni­ties right away. To state the most ba­sic, the ear­lier a sus­pect first ap­pears on the screen, the more likely they are to have done the deed. So the mo­ment one of the sus­pects ap­pears – ‘That’s Bob!’ – ev­ery­one should rush to buy Bob, and per­haps sell ev­ery­one else if trad­ing costs make that a good idea. How far up to buy him, or sell oth­ers, is an open ques­tion.

That will be the first of many times when there will be an ‘ob­vi­ous’ up­date. There will also be non-ob­vi­ous up­dates. Stay­ing phys­i­cally close to the board, chips and/​or con­tracts ready to go, is key to make sure you get the trade first. This im­plies that mak­ing a race de­pend on the phys­i­cal ex­change of items might be a prob­lem. Let­ting it be ver­bal (e.g. who­ever first says ‘I buy Bob’) pre­vents that is­sue, but risks am­bi­guity.

What char­ac­ter­izes the early game, as op­posed to the late game, is that the fo­cus is on ‘make good trades’ rather than on win­ning. There’s no rea­son to worry too much about who owns how many of each con­tract, un­less some­one is in­vested heav­ily in one par­tic­u­lar sus­pect. We can think of that as a player choos­ing to en­ter the endgame early.

### Attention

Robin notes an in­ter­est­ing phe­nomenon, that play­ers got caught up in the day trad­ing and ne­glected to watch the mys­tery. Where should the smart player di­rect the bulk of their at­ten­tion?

That de­pends upon your model of mur­der mys­ter­ies.

One model says that mur­der mys­ter­ies are ‘fair’. Clues are in­tro­duced. If you pay at­ten­tion to those clues, you can figure out who did it be­fore the de­tec­tive does. When the de­tec­tive solves the mys­tery, you can ver­ify that the solu­tion is cor­rect once you hear their logic. If you can solve the mys­tery first, you can sell ev­ery worth­less con­tract and buy all the worth­while con­tracts. Ideally, that should be good enough to win the game, es­pe­cially if you ex­e­cute prop­erly, sel­l­ing and buy­ing in bal­ance with­out giv­ing away that you be­lieve you’ve solved the mys­tery.

Another re­lated model says that mur­der mys­ter­ies fol­low the rules of mur­der mys­ter­ies, and that this of­ten is good enough to nar­row down or iden­tify the kil­ler. That way-too-fa­mous-for-his-role ac­tor is ob­vi­ously the kil­ler. Another would-be sus­pect was in­tro­duced at the wrong time, so she’s out. A third could eas­ily have done it, but that wouldn’t work with the the­matic el­e­ments on dis­play.

A third model says that the de­tec­tive, or oth­ers in the movie, have a cer­tain cred­i­bil­ity. Thus, when Sher­lock Holmes says that Bob is in­no­cent, that is that. Bob is in­no­cent. You don’t need to know why. Ev­i­dence oth­er­wise might not mean much, but there’s some­one you can trust.

Func­tion­ally, these three are iden­ti­cal once you know what fac­tors you’r re­act­ing to. They say that (some kinds of) ev­i­dence count as ev­i­dence, and re­sult­ing up­dates are real. The more you be­lieve this, the more you should pay at­ten­tion to the movie. This in­cludes try­ing early on to figure out what type of movie this is. Be Genre Savvy! Un­til you know what rules ap­ply, don’t worry too much about day trad­ing, un­less peo­ple are go­ing nuts.

A fourth model says that the mys­tery was cho­sen for a rea­son, and writ­ten to keep up sus­pense, so noth­ing you learn mat­ters much be­yond es­tab­lish­ing who the sus­pects are. The game already did that for you. Un­less the game fol­lowed my ad­vice and in­cluded an ob­vi­ously fake sus­pect or two, to pun­ish play­ers who only look at trad­ing.

If you be­lieve this model, and don’t think there is news and there aren’t fake sub­jects (or that they will be suffi­ciently ob­vi­ous you’ll know any­way, if only by how oth­ers talk and act) then you won’t put as much value on watch­ing the movie. If trad­ing has good ac­tion, trad­ing might be a bet­ter bet.

A fifth model that can over­lap with the pre­vi­ous mod­els says that oth­ers will watch the movie and pro­cess that in­for­ma­tion, so there’s no need to watch your­self if there are enough other play­ers. You might think that there is then a mo­men­tum effect, where play­ers are un­will­ing to trade ag­gres­sively enough on new in­for­ma­tion. Or you might think that play­ers over­re­act to new in­for­ma­tion, es­pe­cially if you’re a forth-model (noth­ing mat­ters, eat at Arby’s) kind of the­o­rist.

If you feel oth­ers can be re­lied on to re­act to news, you might trade on news even if you don’t think it mat­ters, be­cause oth­ers will trade af­ter you, and you can then cash out at a quick profit. Just like in the real mar­kets.

Or you might con­cen­trate on ar­bi­trage. Robin ob­served that play­ers would fo­cus on buy­ing good sus­pects rather than sel­l­ing poor sus­pects, and this of­ten re­sulted in prob­a­bil­ities that summed to more than 100%. This offers the chance for risk-free prof­its, plus the chance to place what­ever bet you like best while cash­ing in.

In my mind, the ques­tion boils down to where the game will be won and lost. Is there enough profit in day trad­ing to beat the per­son who placed the largest bet on the guilty party? What hap­pens in the endgame?

### The End Game

A player en­ters the endgame when they at­tempt to en­sure that they win if a par­tic­u­lar sus­pect is guilty.

This is not as difficult as it looks, and could be quite difficult to fight against. Sup­pose I want to bet on Alice be­ing the culprit. I could sell all other sus­pects and buy her con­tracts. As a toy ex­am­ple, lets say there are four sus­pects, and lets say I de­cide to butcher my ex­e­cu­tions. I sell the oth­ers for \$20 and \$15 each, and buy Alice for \$25, \$30 and \$40.

If the game ends and Alice is guilty, I made \$105 sel­l­ing worth­less con­tracts, and made \$195 buy­ing Alice con­tracts, for a net profit of \$295. If she’s in­no­cent, I col­lect noth­ing, so I paid \$105 for Alice con­tracts and made \$105 sel­l­ing other con­tracts, so I’m just out my ini­tial \$200 and die broke. That’s re­ally, re­ally ter­rible odds if I chose Alice at ran­dom!

But if it’s a 10 per­son game and I do that, even if I chose at ran­dom, 25% of the time Alice is guilty. Can some­one else make more than \$295 to beat me?

If af­ter I finish, oth­ers re­turn all the prices to nor­mal, then some­one else could profit from my ini­tial haste, then ex­e­cute the same trades I did at bet­ter prices. If that hap­pens, I’m shut out.

That works if you jump the gun, and en­ter the endgame too early. That’s true even if Alice is the most likely sus­pect.

In par­tic­u­lar, oth­ers now need to make a choice. Lets say I went all in on Alice. There are three ba­sic ap­proaches on how to re­spond:

1. Aban­don Alice. If Alice is guilty, you’ve lost. So it’s safe to as­sume Alice is in­no­cent, and sell any Alice con­tracts at their new higher prices, es­pe­cially once you’re broke and no longer can buy any more. If this en­courages some­one else to take op­tion 2 and also move in on Alice, even bet­ter, that’s one less per­son who can beat you if Alice is in­no­cent. The ma­jor­ity of play­ers should do this.

2. At­tack Alice. If oth­ers are aban­don­ing Alice in droves, her price might col­lapse even be­yond \$25 as peo­ple rush to sell. You can then pick con­tracts up cheap, sell other con­tracts at bet­ter prices, and have a strictly bet­ter po­si­tion.

3. Ar­bi­trage. Try to make as much money off the situ­a­tion as pos­si­ble, with­out com­mit­ting to a di­rec­tion. If peo­ple are be­ing ‘too strate­gic’ and too ea­ger to get where they want to go, rather than fo­cus­ing on get­ting the best price, then by mak­ing good trades (sell Alice when I buy too quickly, buy when oth­ers sell too quickly) and forc­ing oth­ers to get worse prices, I can end up with more value, then de­cide later what to do.

If you only en­gage in ar­bi­trage, and oth­ers com­mit to sus­pects, you’ll be in a lot of trou­ble un­less you’ve already made a ton, be­cause you won’t have any­one to trade against. Your only op­tion be­comes to trade with the mar­ket, which limits how much you can get on the sus­pect you fi­nally de­cide to go with, even if the mys­tery is solved while the mar­ket is open, and you’re the only one left with cash.

The good and bad news is that’s un­likely to hap­pen, as oth­ers will also ‘stick around’ with flex­ible port­fo­lios. That means that you won’t be able to make that much when the mys­tery gets solved, but it does mean you can di­vide the spoils. If six play­ers com­mit to sus­pects while four make good trades, not only are two of the six already shut out, it’s likely the re­main­ing four can co­or­di­nate (or just do sen­si­ble trades) to win if two or three of the sus­pects are found guilty, and some­times should be able to nail all four.

When you have four (or six) sus­pects and ten play­ers, there are not enough sus­pects for ev­ery­one to own one, and there cer­tainly aren’t enough for any­one to own two. That means that even if a sus­pect looks likely to be guilty, if you know you can’t win that sce­nario, you’ll be dump­ing, and that means at least seven of ten peo­ple are dump­ing any given sus­pect if they un­der­stand the game.

The log­i­cal re­sponse to this is to stay far enough ahead on your sus­pect that you clearly win if they’re guilty, and if you get a good early op­por­tu­nity to dump other con­tracts you should definitely do that. Good trades are gen­er­ally good, and those trades just got even bet­ter, es­pe­cially if ev­ery­one fo­cuses on buy­ing rather than sel­l­ing. What you don’t want to do is over­pay, or run out of cash (and/​or run out of things you can sell).

Thus, I might buy the Alice \$20, \$25 and Alice \$30 con­tracts, and start sel­l­ing con­tracts on sus­pects I think are trad­ing rich. What I’m wor­ried about is com­pe­ti­tion – I don’t want other play­ers buy­ing Alice con­tracts, so if they do, I’ll make sure they don’t get size off by buy­ing at least at their price, and I’ll make sure to stay ahead of them on size. I’ll also think about whether the re­main­ing play­ers are so­phis­ti­cated enough to sell what they have, even at lousy prices; if they are, I’ll be care­ful to hold a bunch of cash in re­serve. If there are ten play­ers, I can ex­pect there to ex­ist 16-25 Alice con­tracts, and I want to be sure not to run out of money.

### Rank Ordering

This sug­gests each player has a few differ­ent goals.

You want to ac­cu­mu­late con­tracts in sus­pects you ‘like’ (which mostly means the ones you think are good bets), so you can get ‘con­trol’ of one or more of them. Con­trol means that if they did it, you win.

You want to get rid of con­tracts in the sus­pects you don’t like. The trick here is that some­times the price will go su­per high (rel­a­tive to the prob­a­bil­ity they did it) as mul­ti­ple play­ers com­pete to gain ‘con­trol’ of the sus­pect. Other times, the price will col­lapse be­cause there is only one bid­der for con­trol of that sus­pect. If one player gets a bunch of con­tracts, and is in good over­all shape, then no one else will com­pete.

That in turn might drive the price so low – \$10 or even \$5 – that the value of their port­fo­lio shrinks a lot, tempt­ing an­other player to en­ter, but do­ing so would drive the price up right away, so it of­ten doesn’t make sense to com­pete. If Bob is buy­ing up Alice con­tracts and Carol now buys one at \$10, who is go­ing to sell one now? Much bet­ter to wait to see if the price goes higher, which in turn puts Bob back in con­trol. The flip side of that is, if Carol can buy a \$10 and a \$15 con­tract, and force Bob to then pay \$20, Carol can sell back to Bob at a profit. It’s a risky bluff if oth­ers are ac­tively sel­l­ing, but it can definitely pay off.

The key in these fights is who has more over­all port­fo­lio value, plus the trans­ac­tion costs of mov­ing into more con­tacts. If Carol can make \$100 trad­ing back and forth in other con­tracts, Bob is go­ing to have a tough time keep­ing con­trol, and mostly has to hope that Carol chooses to go af­ter a differ­ent sus­pect. By be­ing in as good shape as pos­si­ble, Bob both is more likely to win the fight, and (if oth­ers re­al­ize this) more likely to avoid the fight.

With a lot of play­ers en­gaged in ac­tive day trad­ing, and aren’t strate­gi­cally fo­cused, trans­ac­tion costs could be low. If they’re suffi­ciently low, then it could be a long time be­fore it is hard to buy and sell what you want at a rea­son­able price, post­pon­ing the end game un­til quite late. The more other play­ers are strate­gi­cally fo­cused, and strat­egy de­ter­mines price, the harder it is to trade, the more ex­ist­ing po­si­tion­ing mat­ters and the less you can try to day trade for profit other than an­ti­ci­pat­ing a fight over a sus­pect, or a dump of them.

### Rich Player, Poor Player

Sup­pose you’re a poor player. You made some trades, and they didn’t work out. Per­haps you held on to a sus­pect or two too long, and oth­ers dumped them, ei­ther strate­gi­cally or for value. Per­haps you had a hunch, got overex­cited, and oth­ers dis­agreed, and now you’re look­ing fool­ish. Now you only have (let’s say) \$120 in equity, down from \$200.

You Play to Win the Game. How do you do that? There are more play­ers than sus­pects, sev­eral of whom have dou­ble your stake. So you’ll need to find a good gam­bit.

A ba­sic gam­bit would be to buy up all the con­tracts you can of a sus­pect ev­ery­one has dis­missed. Even if there are very good rea­sons they seem im­pos­si­ble and are trad­ing at \$5, you can still get enough of them to win if it works out, and you might have no com­pe­ti­tion since play­ers in bet­ter shape have juicier tar­gets. Slim chance beats none.

But if even that’s out of the ques­tion, you’ll have to re­build your po­si­tion. You will need to spec­u­late via day trad­ing. Any other play locks in a loss. You find your­self in a hole, and have no choice but to keep dig­ging. Small ar­bi­trage won’t work. Your best bet is likely to watch the screen and listen, and try to re­act faster than ev­ery­one else in the hopes that the lat­est de­vel­op­ment is huge or seen as huge, then turn around and sell your new po­si­tion to oth­ers to make a quick buck. Then hope there’s still enough twists to do this again.

If the endgame has ar­rived, and rich play­ers are sit­ting on or fight­ing for all the sus­pects, you’ve lost. Your best bet is to con­soli­date into cash, and hope some sus­pect crashes down to \$5 for you.

Now sup­pose you’re a rich player. You have \$300 in equity. How do you max­i­mize your chance of win­ning?

The ba­sic play is to cor­ner the mar­ket on the most likely sus­pect, or who­ever you think is most likely. If you make a strong move in, you should be able to scare off com­pe­ti­tion, and even if you don’t do so, you can use that as an op­por­tu­nity to make more profit if they drive the price up. At some point, oth­ers will have to dump, and you can af­ford to give them a good price if you have to. It’s hard to win a fight when out­gunned. The key is not to en­gage too much too soon, as this risks let­ting a third player take ad­van­tage of an as­set dump later. So you’ll want to hold some cash for that, if pos­si­ble. Re­mem­ber that you’ll need some­thing like 12-14 con­tracts to feel safe from a dump, de­pend­ing on how much equity you’ve built, if you’re out of cash. That shuts out other play­ers.

The ad­vanced play, if you’re suffi­ciently far ahead, is to try to win on mul­ti­ple sus­pects. That’s su­per hard. Even if you had \$400 in equity, if you di­vide it in half, there are still mul­ti­ple other play­ers over \$200. It seems un­likely you can get con­trol of mul­ti­ple worth­while sus­pects. There’s no point in try­ing for mul­ti­ple bar­gain base­ment sus­pects at the ex­pense of one good one, even if it works. So is there any hope here?

I think there is, in the sce­nario where there is a clear prime sus­pect.

In this sce­nario, the prime sus­pect was bid high early on. Given Robin’s notes about player be­hav­ior and ten­dency to push prices too high, and the bat­tle for con­trol of the sus­pect, prices might get very high very quickly. There also may be play­ers who will re­fuse to sell their con­tracts in the prime sus­pect, be­cause they don’t re­al­ize that they’re shut out of win­ning in that case. Either they’re max­i­miz­ing ex­pected value rather than chance of win­ning, or they don’t re­al­ize the prob­lem, or both.

This could open up an op­por­tu­nity where the ‘net profit’ on the prime sus­pect isn’t that high for any player. Sup­pose they start at \$25, and ev­ery­one starts with their two con­tracts. They then trade at \$30, \$40, \$50 and \$60 in a row, not all to the same player. So there’s min­i­mal chances to buy con­tracts that make you that much money. If you buy an \$80 con­tract your max­i­mum profit is \$20, which is easy to beat by day trad­ing.

So what you can do is go for the block. Hope­fully you helped drive the price up early, which is part of how you got your equity. Then con­tracts only re­ally traded at \$60 and \$80. So even if the sus­pect is guilty, some­one who moved in on this with­out day trad­ing first is not go­ing to end up with many con­tracts. You start with \$200, so lets say they end up with 3 con­tracts and a lit­tle cash.

It’s not crazy for you to sell the sus­pect at the top, do some suc­cess­ful day trad­ing, and then have over \$300 in cash. You could win with­out any con­tracts that pay out, if you know you’re the most suc­cess­ful day trader and no one can have that many con­tracts.

That’s a bet­ter po­si­tion than hav­ing 5 or 6 con­tracts in the prime sus­pect, since you still have cash if they’re in­no­cent. The trick is then hav­ing that be enough to win on an­other sus­pect as well, or split­ting your efforts by hold­ing onto con­tracts el­se­where. Tricky. But per­haps not im­pos­si­ble, es­pe­cially if peo­ple are dump­ing con­tracts at \$5.

At a min­i­mum, what you can do is be in a strong po­si­tion to re­spond to new de­vel­op­ments, and be able to choose which other sus­pect to back later in the game if you now think they’re more likely, while still win­ning if the situ­a­tion doesn’t change. That’s very strong.

A fi­nal note is that it is le­gal, in the game, to trade with an­other player with­out go­ing through the mar­ket. This could be used to buy out a play­ers’ po­si­tion in a sus­pect, shift­ing con­trol of that sus­pect, and avoid the is­sue where once a player starts dump­ing a po­si­tion, the price will col­lapse, as well as the abil­ity of other play­ers to ‘in­ter­cept’ the trans­fer and ruin the buy­ers’ at­tempt to ac­cu­mu­late a new po­si­tion and take con­trol. Thus, play­ers should learn that if they have a bunch of con­tracts and want out, they should check for a bulk buyer, and if they want in they should con­sider do­ing the same. The risk of course is that you tip your hand, which makes do­ing it on the buy side less at­trac­tive.

### Flex­ible Structures

It’s also worth not­ing that you can ex­tend the idea eas­ily to other pre­dic­tion mar­kets, and to an on­line ver­sion.

You could trade on the out­come of a sport­ing event, or an elec­tion, or any other real-world pre­dic­tion mar­ket, us­ing the same rules. You could play a board game, and also play the con­tract game on the out­come of your board game. That gives play­ers some­thing to do be­tween turns and ex­tra things to think about, and gives ex­tra play­ers or elimi­nated play­ers some­thing to do.

You could trade over a se­ries of out­comes or events (for ex­am­ple, all the foot­ball games played to­day, or both the win­ner of the game and the com­bined num­ber of points scored, or even ob­scure stuff too like num­ber of punts or what not) in or­der to re­ward more trad­ing ‘for value’ and place less em­pha­sis on be­ing right. Or just keep track of funds be­tween games, watch mul­ti­ple shows, and re­ward the over­all win­ner.

That raises the ques­tion of what we can learn about pre­dic­tion mar­kets from the game.

### Mar­ket Ac­cu­racy and Implications

Early in the game, mar­ket prices should roughly re­flect fair prob­a­bil­ities of be­ing guilty. Any­one who jumps the gun for strate­gic po­si­tion­ing will lose out to a more pa­tient player. That won’t stop play­ers from be­ing overea­ger, and bid­ding sus­pects up too high, but as Robin noted that opens the door for oth­ers to do ar­bi­trage and sell the con­tracts back down to rea­son­able prices.

Later in the game, prices will grow in­creas­ingly in­ac­cu­rate as play­ers jockey for po­si­tion, and let strate­gic con­sid­er­a­tions over­ride equity con­sid­er­a­tions.

This is a phe­nomenon we see in many strate­gic games. Early in the game, play­ers mostly are con­cerned about get­ting good value, as most as­sets are vaguely fun­gible and will at some point be use­ful or nec­es­sary. As the game pro­gresses, play­ers de­velop spe­cial­ized strate­gies, and start to do what­ever they need to do to get what they need, of­ten at ter­rible ex­change rates, and dump or ig­nore things they don’t need, also at ter­rible ex­change rates.

If we wanted to im­prove ac­cu­racy, we’d need to make the game less strate­gic and more tac­ti­cal, by re­ward­ing play­ers who max­i­mize ex­pected prof­its. There’s a dumb mar­ket that is hand­ing out Free Money when news oc­curs. We’d like play­ers to bat­tle for a share of that pie, rather than com­pet­ing for con­trol of sus­pects. If the game was played over many rounds, the early rounds would mostly fo­cus on ex­pected value and do­ing good trades. If the game was played for real money, and set­tled in ac­tual dol­lars, then we’d definitely have a lot more ac­cu­rate pric­ing!

If a mar­ket has traders purely mo­ti­vated by ex­pected value and profit, then its pric­ing will be as good as the pric­ing abil­ity of the traders.

If a mar­ket has a few ‘mo­ti­vated’ traders, or noise traders, that are do­ing some­thing for rea­sons other than ex­pected value, that is good. You need a source of free money to make the mar­ket work. Thus, the ex­is­tence of the bank, as a source of free money, is great, be­cause it mo­ti­vates the game. You can imag­ine a ver­sion of the game where play­ers can only trade when they agree to it. There would still be trades, since the prize for win­ning should over­come fric­tions and ad­verse se­lec­tion, but vol­ume of trad­ing would plum­met.

If a mar­ket has a few peo­ple who have poor fair val­ues, that works like mo­ti­vated traders.

If a mar­ket has too many traders who have poor fair val­ues, or in con­text they have fair val­ues that are not based on the ex­pected pay­out, then re­la­tion­ships break down. There’s now profit for those who bet against them, but that doesn’t mean there’s enough money in the wings to bal­ance things out. At some point that money is ex­hausted, and no one pay­ing at­ten­tion has spare funds. Prices stop be­ing ac­cu­rate, to vary­ing de­grees.

In par­tic­u­lar, this illus­trates that if those man­ag­ing the money have pay­outs that are non-lin­ear func­tions of their prof­its, then very weird things will hap­pen. If I get fired for miss­ing my bench­mark, and so do my col­leagues, but we don’t get ex­tra fired for miss­ing them by a lot, then this will lead us to dis­count tail risks. In the game, this takes the form of dump­ing sus­pects you can’t con­trol – if Alice did it, you’ve already lost, and third prize is you’re fired. There are many other similar sce­nar­ios. If we want ac­cu­rate prices, we need traders to have lin­ear util­ity func­tions, or rea­son­able ap­prox­i­ma­tion thereof.

### Overall

This game sounds like a lot of fun, and seems to have lots of op­por­tu­ni­ties for deep tac­ti­cal and strate­gic play, for bluffing, and to do things that play­ers should find fun. I re­ally hope that it gets made. You can take one or more of many roles – ar­bi­trager, mys­tery solver, genre savvy lo­gi­cian, mo­men­tum, value, tac­ti­cal or strate­gic, or just hang out and watch the fun and/​or the mys­tery, and if you later get a hunch, you can go for it.

I hope to talk to a few friends of mine who have small game com­pa­nies, in the hopes one of them can help. Kick­starter ho?

If any­one out there is in­ter­ested in the game and mak­ing it hap­pen, please do talk to Robin Han­son about it. I’m sure he’d be happy to help make it a re­al­ity. And if you’re look­ing to play, I en­courage you to give it a shot, and re­port back.

No nominations.
No reviews.
• Loved this. The vast ma­jor­ity of anal­y­sis on games is shal­low, tend­ing to look at the stated rules and ex­plicit me­chan­ics, and ig­nor­ing de­rived and im­plied rules (the vastly differ­ent prop­erty value-pay­off heatmaps in Monopoly, the “genre aware­ness” here), ig­nor­ing tempo/​timing is­sues, and ig­nor­ing win con­di­tions /​ endgame con­sid­er­a­tions.

I love when stuff like this gets boiled down el­e­gantly:

>You want to ac­cu­mu­late con­tracts in sus­pects you ‘like’ (which mostly means the ones you think are good bets), so you can get ‘con­trol’ of one or more of them. Con­trol means that if they did it, you win.

Ah, cool, that’s a win con­di­tion.

And then the log­i­cal corol­laries:

>… Sup­pose you’re a poor player. … A ba­sic gam­bit would be to buy up all the con­tracts you can of a sus­pect ev­ery­one has dis­missed. Even if there are very good rea­sons they seem im­pos­si­ble and are trad­ing at \$5, you can still get enough of them to win if it works out, and you might have no com­pe­ti­tion since play­ers in bet­ter shape have juicier tar­gets. Slim chance beats none. But if even that’s out of the ques­tion, you’ll have to re­build your po­si­tion. You will need to spec­u­late via day trad­ing. Any other play locks in a loss.

And tempo:

>This is a phe­nomenon we see in many strate­gic games. Early in the game, play­ers mostly are con­cerned about get­ting good value, as most as­sets are vaguely fun­gible and will at some point be use­ful or nec­es­sary. As the game pro­gresses, play­ers de­velop spe­cial­ized strate­gies, and start to do what­ever they need to do to get what they need, of­ten at ter­rible ex­change rates, and dump or ig­nore things they don’t need, also at ter­rible ex­change rates.

I love this stuff. It’s so cool. Hat’s off.

In­ci­den­tally, did you ever read “The best Puerto Rico strat­egy”?

A gem of a broad overview that similarly looks in depth across the whole stack. Puerto Rico is a won­der­ful game in that it’s in­cred­ibly sim­ple and satis­fy­ing for new play­ers — no overtly de­struc­tive ac­tions against fel­low play­ers, ev­ery­one gets a turn, you’re always build­ing up your is­land no mat­ter what — but has in­cred­ibly deep math­e­mat­i­cal and be­hav­ioral com­plex­ity un­der­neath the hood. Rare that games can hit both of those notes.

Definitely recom­mend that one if you haven’t read it. Se­ri­ously, huge re­spect and hat’s off for this ar­ti­cle — be­ing able to in­tuit game dy­nam­ics, strate­gic and tac­ti­cal con­sid­er­a­tions, elu­ci­dat­ing win con­di­tions and rele­vant play ad­just­ments when win­ning or los­ing be­fore hav­ing played the game ex­ten­sively is… damn im­pres­sive.

I’m to­tally hop­ing you teach a class on game anal­y­sis at Stan­ford or MIT or what­ever some­day, and put the lec­tures on­line. I’d watch ’em. What you do is, frankly, re­ally damn im­pres­sive. No idle flat­tery, but where do you reckon you’re at per­centile-wise in this skill? Top 0.00001% of an­a­lysts/​the­o­rists would be 1 in 10 mil­lion or so. There’s prob­a­bly not more than 30 bet­ter an­a­lysts in the USA than you, no? No flat­tery, more like a fac­tual state­ment. If some­one gave me an un­der/​over bet of “100 more tal­ented game an­a­lysts than Zvi in the USA”, I’m lean­ing su­per strongly to the the un­der; un­der/​over of 1000 peo­ple and I take the un­der with­out hes­i­tat­ing.

• That’s some strong praise there. It’s great to hear and I hope I can live up to it. I think I’m one of the strongest at some as­pects of anal­y­sis, and this task here plays into a lot of my strengths in­clud­ing my trad­ing and mar­ket mak­ing ex­pe­rience. In other ways, I’m not as strong. I

re­ally en­joy do­ing this type of anal­y­sis not only on games but on real world situ­a­tions, prob­lems, mechanisms, busi­ness op­por­tu­ni­ties, and so forth. If I could get fairly com­pen­sated for that type of con­sult­ing I’d love to do it, but alas the con­sult­ing busi­ness is mostly a self-sel­l­ing busi­ness and the in­ter­nal cor­po­rate poli­tics busi­ness as far as I can tell—you get ad­vice like “never im­prove things more than 10%, and if you do im­prove it more make sure to hide it.”

In much more promis­ing news, I’m ex­plor­ing a po­ten­tial game de­sign op­por­tu­nity, but it’s too early for me to say more than that yet.

I’ll check out the guide, looks cool at first glance.

• Note that all this anal­y­sis is based on think­ing about the game, not from play­ing the game. From my ob­serv­ing game play, I’d say that price ac­cu­racy does not no­tice­ably suffer in the endgame.

For game de­sign, yes good to in­clude a few char­ac­ters who will be ex­cluded early, so peo­ple at­tend to story in early pe­riod.

• What types of play­ers did you test the game on, and how many games did they each play?

I can think of many other games where this dis­tor­tion effect doesn’t hap­pen with new play­ers, as they don’t think about the game end­ing or the strate­gic layer, then picks up as play­ers gain ex­pe­rience and im­prove. So this re­sult isn’t that sur­pris­ing for play­ers on their first game, es­pe­cially if they’re not hard­core game play­ers. But it would be sur­pris­ing if it was a sta­ble equil­ibrium.

• Yes, we only did a half dozen tri­als, and mostly with new play­ers, so play­ers were in­ex­pe­rienced.

• I might be miss­ing some­thing, but I got the im­pres­sion that your in­ter­pre­ta­tion is, it’s an all-or-noth­ing game where the win­ner is the per­son who made the most money. But the more nat­u­ral in­ter­pre­ta­tion seems to be: your util­ity func­tion is just the amount of money you made. So, it’s not im­por­tant whether you made more than an­other player, only the ab­solute amount is im­por­tant. The lat­ter is more in line with simu­lat­ing pre­dic­tion mar­kets (and is most eas­ily achieved if the game is on real money).

• Robin ex­plic­itly said “the per­son with the most money wins” and that’s the most nat­u­ral way of view­ing it as a game. Of course, there’s noth­ing *wrong* with do­ing it the other way, and it cre­ates more ac­cu­rate (re­al­is­tic?) prices and mar­kets, as I note. But it’s im­por­tant to note that *as a game* it’s more in­ter­est­ing to try and get the most money, than it is to sim­ply make good trades. If it’s nor­mal trad­ing you’re all tac­tics and no strat­egy. This way you get both, plus the thrill of vic­tory and the agony of defeat.

• Oh, now I see that you do men­tion it. I missed that part, my apolo­gies.

• I won­der if some­thing like this could be used in the work­place.

If we get an un­ex­pected test re­sult we all get to­gether and come up with hy­pothe­ses as to what might be the cause. Each hy­poth­e­sis has a column plus an “other” column and the game be­gins.

You’d need a ro­bust method of adding new hy­pothe­ses (pos­si­bly which re­warded the hy­poth­e­siser—maybe giv­ing them the first chance to pur­chasie stocks), plus a way of mak­ing it fair when new ex­per­i­men­tal ev­i­dence comes in (not just first come first served!).

I re­al­ise this is es­sen­tially just a pre­dic­tion mar­ket at work but I think the vi­sual el­e­ments would help peo­ple get it even if they aren’t fa­mil­iar with the the­ory. I imag­ine it would help with con­sid­er­ing what to test next and to make sure peo­ple keep an open mind.

• I’d love to know what else this could work for. Slasher flicks might work, or any movie with an en­sem­ble cast that gets kil­led off one by one un­til only one or two re­main. Play­ers would prob­a­bly have to know in ad­vance ex­actly how many sur­vive, such that the ini­tial trad­ing val­ues can be set ap­pro­pri­ately (if 2 of 8 sur­vive, then the ini­tial trad­ing value should start at 30 to buy, 20 to sell.) The num­ber of sur­vivors would also set the cost of the “buy all con­tracts” op­tion.

Slasher flicks would have more ac­tion early on as char­ac­ters are elimi­nated in grue­some fash­ion, but the win­ner might also be de­ter­mined well be­fore the end of the movie.

• You don’t need to know that one and only one con­tract pays. How­ever, if you don’t know that, then you can’t al­low peo­ple to ex­change \$100 for a set of con­tracts (or vice versa). So you could have con­tracts on each per­son sur­viv­ing, al­though you’d need to clar­ify care­fully what did and didn’t count, in ad­vance. And the strat­egy would be differ­ent, since there’s the chance no one sur­vives, or mul­ti­ple peo­ple sur­vive. You could also have con­tracts on other stuff (e.g. who dies first, then sec­ond, etc, or what kills them, or what not).

Although in most slasher movies the vir­gin would just trade at \$90 and ev­ery­one else su­per cheap...

But yeah, the logic ex­pands.

• If more than one per­son “did it”, you could pay off that frac­tion of \$100 to each. So if two did it, each card is worth \$50 at the end.

• I like it, that works well, so long as we have an air­tight defi­ni­tion in ad­vance of when this counts. Alter­na­tively, we can know from our guide that the re­sult won’t be am­bigu­ous.

• That works. Though now card price doesn’t ac­tu­ally re­flect a char­ac­ter’s im­plied prob­a­bil­ity of sur­viv­ing. Eg buy­ing a card at \$40 is a con­fi­dent move if there’s a chance of two sur­vivors, and always loses money if there’s 3.

In­stead it’d be \$100*p(sur­viv­ing|1 sur­vivor) + \$50*p(sur­viv­ing|2 sur­vivors) + \$33*p(sur­viv­ing|3 sur­vivors)… which makes it a lot harder to think about whether to buy or sell. Could make things more in­ter­est­ing though.

• In ad­di­tion to not be­ing able to ex­change \$100 for a set of con­tracts, there’d be some awk­ward­ness over de­ter­min­ing where to set the start­ing trade value. No­tice in the origi­nal ar­ti­cle it’s ini­tally \$15 to sell and \$20 to buy, re­flect­ing the fact that a 6 char­ac­ter mys­tery im­plies a base prob­a­bil­ity of 16.6% for each char­ac­ter. If you naively started a slasher at \$15 sell/​\$20 buy, some­one who be­lieves two sur­vivors is likely would im­me­di­ately bid all char­ac­ters up to \$30 sell/​\$40 buy.

Though you’re right, most slash­ers don’t try nearly as hard to sur­prise you as mur­der mys­ter­ies do. A movie that es­tab­lishes a ro­mance sub­plot 10 min­utes in would im­me­di­ately see those two char­ac­ters bid up to at least \$50.