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A Study in Secret Auction


Imagine you're playing chess, and after a hard-fought opening and midgame, one side has a queen and the other side, two bishops: aside from the kings, there are no other pieces on the board. This setup doesn't make good gameplay per se–it's drawish and the strategy is unclear. But it's so fascinating that it came up in a real game; after all, it's so rare and exotic that it's a surprise it came up–and so it becomes interesting. I call this a study. And it's what happened with Secret Auction: Game 2.


Welcome to Chromatic Conflux! Secret Auction returned for its second game this month, and it's a doozy.* Let's get into the nitty-gritty of the runoff situation.

A giant rock split in half. That's what you get when you search "gem split" on Google Images. Source: Wikipedia.

The Runoff Rules

It all started innocently enough. For variety's sake, I swapped the tie rules from gem split to runoff. Here's how the runoff rules work.


Imagine that Alice, Bob and Carol are playing Secret Auction. Alice and Bob each bid $10 on the 5-point diamond, tied for highest, whereas Carol bids only $5. Under the gem split system, Alice and Bob would each get 2.5 points. Whereas under the runoff system, each player loses their bid, and the game proceeds into an emergency runoff round, titled Round 5B. Note that even Carol can participate in the runoff–it's not just Alice and Bob. In fact, Carol dominates the runoff. She bids $10, which is higher than Alice or Bob could possibly bid, as they each have a measly $5 in their wallet.

It all started innocently enough.

In practice, the runoff essentially takes tying players out of contention for the gem they are bidding for, which is a bit counterintuitive. But in the beginning, I didn't think this all through. I was making a fun change to spice things up.


The Afterthought Clause

Oh, and by the way–you can't have more than four runoffs. If no one has emerged by then, no one wins the gem. So the game doesn't accidentally continue forever.


Regarding $15 Bids

Rounds 1 through 3 were each claimed by one player bidding $15. This is a bit weird from a strategic perspective. If I bid my entire budget on the amethyst, I intend to win its 1 point. Even bidding anything on Round 1 implies that additional points are needed to break ties in the long-term. But fast forward to Round 2. Someone's going to win the round! That person will beat me. Perhaps the idea is that I'll be virtually guaranteed to get points at all, and not to win. That said, to bid $15 on Round 1 only wins if the other rounds deadlock after four runoffs, the afterthought clause I added to the runoff rules. Note that this wasn't true under Gem Split, since the top prizes could be split among many players.


The Runoff Mess

But Round 4A (then called Round 4) was where the runoff mess started. Three players tied for the emerald. Time for Round 4B. As it happened, it was a sixteen-way tie. Not a single person bid a cent. (This suggests a strategy of bidding at least a cent on each round–more on some rounds, obviously.) The situation was resolved in 4C when one player, who we'll call Marcus,** bid $15, earning the emerald. Marcus was now in the lead with 4 points.


Now to the diamond. At this point, five players had cash in the bank. Four had $15 each, and a fifth player had $3. The person with $3 and two of the others decided to bid it all. Another tie.***

The final pawns had been traded off: the game-theoretical fun times were fantabulous.

Don't worry: the quote doesn't make any more sense in context.


The Runoff Study

The final pawns had been traded off: the game-theoretical fun times were fantabulous. Two players, Heliod and Leslawa****, with $15 in Round 5B in a quest to claim the game-winning diamond. Everyone else was bankrupt.


If both players bid it all, the game would go into Round 5C, but no one had any money, so it's a tie. Round 5D. Tie. Round 5E. Tie. That's the last runoff–remember the afterthought clause–so Marcus ends up winning! If both players tied with $0 (or any other amount), then the situation would have been essentially repeated, but this could only happen a few times, at which point Marcus wins.


Heliod's path to victory was two-pronged:

(a) Convince Leslawa not to bid $15.

(b) Bid $15.


This is somewhat awkward, since Leslawa's path to victory was the opposite:

(a) Convince Heliod not to bid $15.

(b) Bid $15.


Part (a) is hard, so the natural game plan was for each player to bid $15. And that's exactly what happened in the game. But the two finalists could have done better.

An animation for the Prisoner's Dilemma. Source: This Place (Youtube).

The situation Heliod and Leslawa are in is akin to the Prisoner's Dilemma. Cooperating is bidding $0 and defecting is bidding $15. If we both defect, we both lose. If I defect and you cooperate, I win. Normally in the Prisoner's Dilemma, if we both cooperate, we both win to a lesser extent. In this case, it's a sort of purgatory where it sends the game to repeat it all over again. Cooperating is unsustainable.


In fact, if Leslawa bids $15, Heliod can't win. No matter what he decides to bid. If he bids $0, Leslawa wins. $7.50? Leslawa wins. $14.99? Leslawa wins. If he decides to bid $15, it's a tie, and then Leslawa doesn't win–Marcus does. Clearly, she is in the same predicament–Heliod's bidding $15 makes her loss certain. In game theory terms, if either player bids $15 makes the situation a Nash equilibrium.***** What this means is that if he bids $15, he has no reason to regret it. But what this also means is that if she bids $15, Heliod has no reason to regret his choice, whatever it may be.


Heliod and Leslawa can exploit this property and work together. (Note that colluding is legal and has always been legal in Secret Auction and Avocado vs. Cucumber.) To increase their chances of winning, they agree to a coin flip.****** If it lands heads, he will bid $15 and she $0. If tails, he will bid $0 whereas she will bid $15. And remember–the coin flip's loser won't be inclined to renege on the deal, since their opponent bidding $15 will force them to lose no matter what. Heliod, as well as Leslawa, has upped his victory chances from next to nothing to 50%.


That's beauty right there, by the way. The ideal strategy in this situation is to collude and throw their decisions up to fate.


The Merits of Studies

I'm not sure studies make for the most interesting games practically. Many players felt left out of the last few rounds as they lost their money earlier on. Additionally, it was more annoying on my part as I didn't know how many rounds there would be. But studies are more interesting to analyze. Out of the study was born a rich tale of Heliod and Leslawa!


For the postgame survey, I polled the question of whether to use the runoff system or the gem split system in breaking ties. Fittingly, the responses were a tie. So I'm breaking the tie myself.

For the postgame survey, I polled the question of whether to use the runoff system or gem split system in breaking ties. Fittingly, the responses were a tie.

The Merits of Games

Studies are fun when they come about, but overall, I believe that true games are more fun to play practically, with complexities and uncertainties. They're more user-friendly, and more engaging. This isn't to say I don't like studies. Avocado vs. Cucumber is a study in many ways. Anyway, my apologies to the runoff people. Gem split it is.


In Other News

I'm making three other minor changes to Secret Auction that are unrelated to the main theses of this post, so I'll write them here.


1. I'm eradicating style points. When I last asked about style points in the survey, back in Avocado vs. Cucumber, only two people had an opinion, both in favor, so I let the style points remain. Nowadays, they're weird and confusing, especially in Secret Auction, so I'm removing them for good.


2. For the sake of experimentation and trying new things, I'm adding a bid cap for the next game of Secret Auction. It's now against the rules to bid more than $7.50 on a single gem. Hopefully, along with the reintroduction of the gem-splitting system, this will allow greater strategic diversity, greater fun, and greater participation. But if it doesn't work, we can return to the current system.


3. Finally, the next game will be conducted all at once, i.e. each player submits their bid for each round to me at the very beginning. Therefore, bids in Round 3 will not be affected by bids in Round 2. On the other hand, bidding will be more streamlined, faster, and more convenient. This is not meant to be a permanent change. It's designed to create a faster and easier game of Secret Auction: now with five times more secret.

Remember to study–but also to have fun.

That's all. Remember to study–but also to have fun.

–beautifulthorns


*I didn't know what doozy meant until I just looked it up a minute ago. It felt appropriate here even before I knew what it meant. Merriam-Webster defines a doozy as "an extraordinary one of its kind," which fits.


**The reason I'm giving Marcus a name will become apparent later. Anyway, his name comes from a random name generator. I could use Alice, Bob, Carol, etc., but I want to add more personality.


***The default bid had moved from "$0" to "All your money," so that may have played a part.


****Starting to regret the random name generator. (I only used the name generator for Leslawa–I named the other player Heliod to make a pun basically none of you will get. Those of you who get it will not laugh, because it is not funny.


For what it's worth, the name generator preferred Brita.)


*****A Nash equilibrium is an outcome of a game or scenario where neither player has a reason to change their strategy in light of what the opponent does. For instance, imagine that you and I both want to open up an Italian restaurant on the same street. If one of us opens a restaurant, that one person will get 200 customers. However, if we both open one, we will each get 100 customers. Assuming that we are both trying to maximize how many customers we get, the Nash equilibrium is that we both open our restaurant. Yes, I would have preferred you not open a restaurant. But I can't control what you do. All I know is that I'm getting 100 customers compared to 0.


Note that sometimes, there are multiple Nash equilibria. I could choose to either go home or go to your house, and you have the same situation. Say that we would both be happier if we were at the same place, whether my house or yours. You coming over and me staying home is a Nash equilibrium. So is me heading to your place and you staying put. In either case, no one has any regrets.


******Most likely, this would be orchestrated through Google Hangouts. I believe typing /roll 1d2 gets the job done.

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