kaashif's blog

Programming, with some mathematics on the side

Valuing converters in Sidereal Confluence

2023-08-08

(or: How I learnt to stop worrying and love opportunity cost)

I'm a fan of the board game Sidereal Confluence. It's a game where:

• There are resources: small cubes, large cubes, ultratech (octagons)

• You have converters that change sets of cubes into others (e.g. 2 white and one blue cube into 3 black cubes)

• Your goal at the end of the game is to have the most victory points (VPs). Some converters make VPs, during the game you can research techs that give you VPs, and all resources you've accumulated convert to VPs at set rates.

The meat of the game is trading - players can trade anything, with any terms. The only rule is that trades are binding - you must honour your agreements. This leads to simple agreements like "I'll give you one white cube for one green cube", where the trade is obviously fair. You also sometimes get harder to value trades, like "This turn you give me one small white cube and next turn I'll give you one large black cube".

The rules have a guideline - three small cubes are worth two large cubes, which is worth the same as an ultratech. This helps with valuations.

One of my friends executed a strategy I found interesting:

1. He played the Eni Et, a race which has special converters. For example, a usual converter might take 3 small cubes and give 4 small cubes, for a ratio of 4/3, but an Eni Et converter might have a ratio of 2 or 3.

2. The catch is the Eni Et can't use their special converters and must trade them to other people who can.

3. He sold them permanently early on, valuing a 2 to 4 or 8 to 13 converter pretty highly. After all, they'll make dozens of cubes for you over the game, right?

4. He won. Did we pay too much for his converters?

Questions I want to answer quantitatively:

• A simple one: if someone gives me a cube this turn in exchange for some number of cubes next turn, how many cubes should they demand?

• A harder one: when buying an Eni Et converter permanently, how many cubes should I pay?

Someone on BoardGameGeek has already done the hard work in this post. I don't know exactly how they did it, but they calculated average converter rates and color correction rates (the rate at which players trade non-convertible for convertible cubes). The reason that's difficult is that converters change as technologies are researched, and when exactly techs are done depends on the players, and which techs are drawn from the deck.

From those values, we can calculate the value of 1 small cube at the end of the game. See that forum post for the details, but the final results that I'll use are the values of 1 small cube at the end of the game on each turn:

Turn Endgame Value per Cube
1 4.64
2 3.80
3 3.08
4 2.45
5 1.91
6 1.45

This means that on turn 1, on average (with average converters and trades), a single small cube will turn into 4.64 small cubes after all 6 turns are over.

We can now answer our first question. Assuming we want to keep our endgame value the same and we're paying 1 small cube on turn 1, we should demand 4.64 cubes of endgame value back on turn 2. On turn 2, each small cube is only worth 3.8 at endgame, so we need 4.64/3.8 = 1.22 small cubes back on turn 2.

We can make a table of the number of cubes we should demand on turn X+1 for a single cube on turn X.

Turn # cubes to ask for next turn
1 1.2198
2 1.2355
3 1.2558
4 1.2815
5 1.3194
6 1.4500

One obvious fact is that the rate increases as the game goes on. This is because converters get better as the game goes on and techs get invented. If the best converter has a 1.5 ratio and someone invents a 1 to 10 converter, obviously you should start asking for more cubes in future trades for next turn - people will be able to pay.

Some conclusions:

• I've played the Im'Dril Nomads a couple of times and they're very dependent on future trades. Usually people settle on a rate of something like 1 small cube this turn for 1 large cube next turn. This is a 1.5 rate, which is better than the "average" rate, strictly speaking.

• These are rules of thumb. If, like the Im'Dril, future trades are crucial to run your high effiency converters, it's fine to give out favourable rates. These calculations assume you make average trades and run an average converter - almost no situations are average.

• People will not generally agree to trades more than 1 turn in the future. Situations change too drastically - techs come out, new converters are invented, colonies are colonized. So more involved yield curve calculations are pointless except maybe if you're playing with bond traders.

Valuing Eni Et converters

Let's say the Eni Et are selling their 2 small cube to 4 small cube converter on turn 1. If you run it on all 6 turns, you'll make 12 small cubes.

The Eni Et will try to pitch that to you. It's really worth 12 cubes they'll say, I'll give you a good price - only 4 cubes! Maybe they'll hold an auction and get a bid of 5. It'll pay for itself quickly and let you do those hard techs they'll claim.

Let's make some assumptions:

1. You manage to run the converter every turn.

2. The converter you forego running is an average converter for that turn.

The average converter efficiency on each turn is:

Turn Average Converter Efficiency
1 1.3
2 1.32
3 1.3
4 1.38
5 1.41
6 1.5

If you run the 2 to 4 converter and forego running an average converter, the number of extra cubes produced as a result of buying the converter is 4 - 2 * average converter efficiency. The cost of foregoing the average converter is the opportunity cost, the cost the Eni Et scammers want you to overlook.

We then multiply the extra cubes produced on each turn by the endgame value of a cube on that turn and sum all future extra cube production to find the value at endgame of a 2 to 4 converter.

Turn Endgame Value per Cube Avg Efficiency Extra Cubes Endgame Value Converter endgame value # cubes
1 4.64 1.3 1.4 6.50 22.73 4.90
2 3.80 1.32 1.36 5.17 16.23 4.27
3 3.08 1.3 1.4 4.31 11.06 3.59
4 2.45 1.38 1.24 3.04 6.75 2.75
5 1.91 1.41 1.18 2.26 3.71 1.94
6 1.45 1.5 1 1.45 1.45 1.00

By endgame the value of a 2 to 4 converter should be obvious. If everyone has a 2 to 3 converter as standard, then you should obviously pay at most 1 cube for a 2 to 4 converter.

The values on the previous turns are much less obvious, at least to me.

On turn 1 you should only pay 4.9 cubes for a 2 to 4 converter that you'll run every turn, even though it's going to produce 24 cubes from 12 you put in.

It gets even worse when you realise you won't always run the converter. If you run the converter on only 4/6 turns, you should only be paying 4/6 * 4.9 = 3.27 cubes for it.

I distinctly remember these converters being sold for far more than that, and not always being run.

Conclusions

Opportunity cost matters.

People pitching complex transactions as "really good deals" are probably tricking you.

A converter that you always run is like a 6 turn bond. You put in the cubes to run the converter once (buy the bond at par), collect fixed payments every turn (coupon payments), and once the 6 turns are over, you keep the cubes you put in (you get the principal back).

This kind of means that when the Eni Et sell you a converter, you're not buying a bond, you're buying the option to buy the bond, you don't get the converter with the initial investment on it already. Well, if you negotiate well you might.

The converter valuation was a discounted cash flow analysis of a security.

Normalizing cube values to endgame values is exactly the same thing as calculating present values, where the present is the end of the game. A cash flow further in the past is worth more, just as a cash flow further into the future is worth less.

Cubes aren't granular enough for every trade to be fair. Those decimal places getting shaved off in every trade are important. People very often trade 2 small cubes for one large, just because trading 1.5 small cubes isn't possible. If you're on the winning side of those trades, you'll be very happy.

The same is true for future trades: if you have the choice between running a 1.3 converter and loaning a cube to someone at a 1.5 rate - loan them the cube! They'll likely think it's a good deal if it lets them run a converter.

Even worse: if you have a pile of cubes waiting for a tech you can't do right away, don't leave it in a pile earning 0%, loan it out! All of these calculations are based on the "average" cube making 30% to 50% per turn!

I wish there were more data on Sidereal Confluence and how pros play it! I mean computer-readable transaction data, full game logs, that kind of thing. But people in real life aren't going to fill out logs and the game is much worse online, so I'm not sure that's ever going to happen.

Unless we get AI to play it or something.