I'll check the current wplace mechanics first, since the rates and costs have changed over time and the math depends on them.The short version: pixels are capped by time, not skill, so most of this is an optimization problem. Real game theory only shows up in pixel wars, at the end.

## The rules I'm using
- New players start with 30 charges, and each charge takes 30 seconds to regenerate. That's 120 charges per hour and **2,880 per day**, a hard ceiling on free pixels.
- Level = floor(pixels^0.65 / 30^0.65), each level adds 2 max charges, and each level pays 500 droplets. You also earn 1 droplet per pixel placed.
- Store: +5 max charges costs 500 droplets, and +30 instant charges also costs 500. The country flag is 20,000 droplets and gives back 10% of charges spent while painting in that country. That's effectively a 1/9 (+11.1%) boost, since the refunded charges get refunded too.

These prices come from late-2025 sources, so check your in-game store. The level formula was verified recently.

## The math
Since pixels are time-limited, the goal is to **waste no regen time** and then squeeze out extra pixels. Compare purchases by pixels gained per droplet:

- **Refill:** 30 pixels once for 500 droplets = 0.06 px/droplet, one time only.
- **+5 capacity:** while your bar overflows overnight (or between check-ins), it adds about 5 px/day forever = 0.01 px/day per droplet. It beats a refill after **6 days**.
- **Flag:** +0.111·N pixels/day, where N is your daily output. It beats capacity once N > ~1,800/day.
- **Capacity stops mattering** at 120 × (your longest offline gap in hours). For 8 hours of sleep that's 960 charges.

The best order is capacity up to that cap, then the flag, then refills with everything left.

Droplet income is front-loaded: the marginal droplets per pixel (pixel plus level-up share) works out to 1 + 10.83·(P/30)^-0.35. That's about 4 per pixel early, 2.3 at level 50, and 1.6 at 100k pixels.

## Simulation
I simulated 90 days assuming 8 hours of sleep, a check-in every 2 hours awake, and 90% efficiency:

| Strategy | Pixels at day 30 | Pixels at day 90 |
|---|---|---|
| Hoard droplets | 29k | 156k |
| Refills only | 40k | 183k |
| Flag first | 51k | 230k |
| **Capacity → flag → refills** | **74k** | **273k** |

Flag-first loses because 20k droplets takes about two weeks to save, and you skip the early capacity gains that compound. Capacity-first hits the daily limit within about two weeks. Leveling also gives free capacity, so you need fewer paid upgrades over time. Events can outweigh everything else: one Halloween event paid 2,000 droplets per pumpkin, about 200k for all 100.

## The actual game theory
- **Droplets and XP pay on placement, not survival.** If you only care about droplets, the dominant strategy is to never waste a charge, and where you place doesn't matter.
- **Art is a war of attrition.** An attacker spends 1 charge to undo 1 pixel, and you spend 1 to repair it, so the side with more total regen wins. That's why alliances and templates matter, though they also create free-rider problems. Since your repairs still pay droplets, being griefed costs you art, not income.
- **Cheap defense:** build in low-traffic areas, or make art people want to preserve.

I'm leaving out alt accounts and bots. They technically work, but they're against the rules and get banned.

If you tell me your sleep schedule and how often you check in, I can plug them into the model.