## Carpe Diem, the Optimization Game: Level 1

Less Wrong contributor owencb links us to a fourteen page paper of his that proposes optimizing your time via a three-resource model. You can find the full document here. Here’s his abstract, which is a good summary:

Abstract: We get lots of opportunities to convert between time and money, and it’s hard to know which ones to take, since they use up other mental resources. I introduce the neutral hour as a tool for thinking about how to make these comparisons. A neutral hour is an hour spent where your mental energy is the same level at the start and the end. I work through some examples of how to use this tool, look at implications for some common scenarios, and explore the theory behind them.

This is a great advancement on the two-resource model of money and time.

In the two-resource model, your time can be traded for dollars, and dollars can be used to buy time, so your task is to figure out the exchange rate and take all offers better than the exchange rate, while declining offers below the exchange rate. If you notice yourself running out of time but not money, or vice versa, you can adjust the exchange rate until the market clears.

In Owen’s three-resource model, you have not only money (M) and time (T) but mental energy (E). Some activities, such as doing your taxes, are taxing, and cost mental energy. Other activities, such as taking a nice walk, are recuperative and recover your mental energy. You can use your recuperative resources to turn time (T) into mental energy (E) as needed, so the exchange rate between T and E can be solved for. Thus, by noting the net gain or loss of E, we can observe what Owen calls the NH, or neutral hour, cost of an activity in time, and compare this to your chosen exchange rate.

He also notes that your current level of energy (E) changes the neutral time cost of activities; being below the threshold needed is very bad, but being too far above that threshold means you’ll burn off energy automatically over time, increasing effective cost, so he recommends keeping your energy level in the optimal range as much as possible, while noting implicitly that having too much energy is a small mistake relative to too little. He also gives the example of a chess tournament that is exhausting today but invigorating in the long term, noting that there are in truth far more than two resources in the territory but defends only putting three resources in his map.

If you are not considering the monetary value of your time at all, using a simple two-factor model is a big win versus not using a model at all. Moving from a simple two-factor model to a three-factor model, even a mostly implicit three-factor model, is also a big win. Adding additional factors would add value, but by itself has rapidly decreasing marginal returns. What is far more valuable is to enrich the structure of play; to change from a framework where the resources are effectively fungible, trade is cheap and availability is predictable, and your goal is reducible to something like “maximize combined net present value of all resources if exchanged into money” to a world where the resources other than money are not fungible, exchange rates and availability of trade fluctuate wildly, resource requirements can change unpredictably, and you can run out of key resources and get bottlenecked.

That’s a lot to do at once, however, so we’ll do that in stages. Each post on this will introduce new complexity and richness to the game, then discuss the strategic implications and how that relates to the real world.

You can then add or remove the relevant resources and complexities from the game, as appropriate to the current optimization problem you face in real life. Your goal should be to play a game that is as simple as possible, but no simpler.

My first version of the basic game got way too complex way too quickly to count as a basic version, as did my second, so I will give you my third attempt, which is as simple as possible while actually allowing trade-offs between the three resources in the basic fashions necessary to have this make any sense, and probably plays a lot better. We want to at least have the following:

Three resources: Money, Time and Energy.

Energy decays over time.

You can trade time to get money, or time to get energy.

You need money and energy to do things.

Carpe Diem (Version 1)

A Game of Solitaire

NOTE: There has been a slight change, the four slot now costs four actions to use rather than five, so the game is somewhat more forgiving and interesting.

You will need a deck of ordinary playing cards, and a way to track money and energy.

Start the game with 3 energy and \$8. The game lasts thirteen days.

Each day you get five units of time and draw four cards. The first card costs one unit of time to use, the second costs two, the third costs three and the fourth costs four actions (NOT all five, this turned out to be too harsh).

Clubs gain you one energy when used, Diamonds net zero energy, Hearts cost one energy, and Spades cost three energy.

If you use a card with a number, you get that much money. You can use the card to keep track until you need change.

If you use a card with a Jack, Queen, King or Ace, you spend \$5 and score 10, 20, 30 or 50 victory points respectively. Use the card to track your points.

You can always use your time to freelance, at \$1 per unit of time.

Or, you can recuperate, and gain 1 energy per unit of time.

Unspent time is lost. At the end of the day, you lose one energy (E1) and must pay \$4 in living expenses; you can’t take actions that would cause you to not have the \$4 or the one energy.

When the first card is drawn for the turn, if you have more energy than that card’s numerical value (Face cards are 10, Ace is 1), lose one energy.

Your score is how many victory points you have when the deck runs out, plus one point per \$1 you have.

My first game things went very well and I scored 371, my second not as well and I scored 332, then the third disaster struck and I got 261. There is a lot of luck in how the cards fall, of course. It’s fun and can get pretty tense, give it a try and let me know what you think!

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### 13 Responses to Carpe Diem, the Optimization Game: Level 1

1. Owen CB says:

Interesting game! I tried a few hands earlier.

I was actually playing a strictly easier version of the game, since I missed the rule about losing energy at the start of the turn depending on the first draw. Despite this, my scores were lower: 283, 227, 284, 289, 314, 297. I often got hit by something like having two aces in the same round so that I couldn’t take both, or on one occasion the ace of clubs in the inaccessible 4th slot on day 1. So I may just have been unlucky — but I may also be bad at this!

I think making the opportunities very chunky like this works well for a game, but is often not such a good model for life (though of course sometimes it is).

For the game, I’d be inclined to make the energy loss depend on the last draw each round rather than the first — at present you are doubly rewarded for drawing a high card first, so this would smooth out a little of the variance.

• TheZvi says:

Good call, no question that losing energy from the one slot is pretty punishing, and this would reduce the luck factor somewhat, so I like the change. I think my first game was exceedingly lucky.

2. janrandom says:

I like it and will try it with my boys. My first game was 339 (after the last day finishing with E0 M19 V320). I think I was lucky that I could get to E>4 M>15 quickly, which seems like the sweet spot.

3. Good info. Lucky me I found your blog by chance (stumbleupon).
I have saved it for later!

• TheZvi says:

Neat. Stumbleupon! Feels like I’ve made the big time from a previous life.

4. PDV says:

I played several hands of this this weekend, and then theorycrafted the best possible score. It is about 411. (94% sure that 411 is achievable, 98% sure that 431 is unachievable, only 44% confident that 411 is the best possible score.) Even that is probably unachievable with optimal play.

(411: Get all AKQ109s, 8s except the spade, the 7s of clubs and diamonds, and the 6 of clubs, all in the 1-2 slot. Almost all remaining time gets you energy. With the 4-slot determining energy loss, that can easily fail to be a factor; if the 1-slot is the punishing one, it’s harder but doable.)

In short: Yes, your first game was very lucky.

This has also got me thinking about “playing to your outs” and how/if that relates to Slack.

• TheZvi says:

Playing to your outs is definitely a thing in Carpe Diem. It’s especially a thing if you have a target you’re shooting for, in which case you may have to assume that certain things fall the way you need them. Late in the game, knowing what’s left in the deck becomes very important, so you can allocate what you need to score the important stuff.

Early on, Slack is a bigger deal than it looks, because when you are out of money/energy and need it for big cards, it really hurts and compounds, and you’ll have plenty of time to spend the money (and energy) well later. Later in the game, it’s a smaller deal than it looks because extra resources expire (almost) worthless.

• Teranis says:

Tried doing the same calculation, but I’ve reached a different score. I want to know if I’ve missed something.
I’m assuming no energy loss for the 1st/4th slot, as with optimal luck I’m pretty sure we can always have a high enough card there.
We have 12 turns to pay 4 money and 1 energy each, meaning 48 money and 12 energy.
All A,K,Q cost 60 money and 9 energy, and give 400 score.
All 10, 9 give 76 money and cost 6 energy.
All 8 except spade give 24 money and cost 0 energy.
7 clubs and diamond give 14 money and 1 energy.
6 of clubs gives 6 money and 1 energy.
Total money: 12. Total energy: -25.
Since we’re only taking from the first and second slot, we have 2 time left per turn, meaning 26 time. 25 goes to energy and 1 to money.
Total score: 400 + 13 money, to a score of 413.
J of clubs is worth 6 score (10 score – 5 money, plus 1 energy which will allow us to trade more time for money).
5 of clubs, 6 of diamond, and 7 of hearts are all worth 6 score by the same token.
8 of spades is worth 5.
Our least valuable card is 9 of spades which is worth a score of 6 too, so we have no better trades to make.
Now we’re only left with the no energy loss assumption as an open question…

• Teranis says:

Forgot the beginning 8 money and 3 energy, meaning we can reach a score 11 points higher. 424.

5. Teranis says: