Unfair Flips is a Steam idle game where you flip coins to earn money, then spend that money on upgrades that make you flip faster, earn more per flip, or increase your win probability. I wanted to know: what’s the mathematically optimal upgrade path?

This post is the story of that analysis — the dead ends, the model that finally worked, some surprising findings, and a real playthrough to verify.

The Setup

The game has four upgradeable parameters:

  • Flip probability p: chance of heads, starts at 20%, upgrades add 5pp each (8 levels, costs 1¢ → $100k)
  • Base worth b: cents earned per head in a streak (4 levels: 1¢, 5¢, 10¢, 25¢, $1)
  • Combo multiplier m: each consecutive head multiplies payout (5 levels: 1×→3.5×, costs 1¢→$100)
  • Flip duration d: time per flip (5 levels: 2.0s→1.0s, costs 1¢→$10k)

The k-th consecutive head in a streak earns \(b \cdot \lceil m^{k-1} \rceil\) cents. At max multiplier (3.5×) the sequence per flip is 1¢, 4¢, 13¢, 43¢, 151¢, 526¢, $18.39, $64.34, $225.19, $788.16 — each flip worth exponentially more than the last.

The win condition: 10 consecutive heads. You accumulate cash from failed streaks and spend it on upgrades.

Dead End: Greedy on Marginal Rate of Return

My first instinct was to greedily pick whichever upgrade maximises \(\Delta(\text{earnings/second}) / \text{cost}\). The problem: it’s myopic. Buying a cheap low-value upgrade can prevent you from saving for a more transformative one sooner. The greedy path kept oscillating between small upgrades instead of building toward a big one.

Dead End: Raw Flip Count

The next attempt tried to minimise total flip count. This ignored flip duration entirely — a 0.5s flip and a 2s flip both count as “one flip” but take very different wall-clock time. Any model that doesn’t account for flip duration is measuring the wrong thing.

What Actually Works: Markov Chain + DP

The correct model treats the game as a Markov chain over streak states \(\{0, 1, \ldots, 9\}\).

Expected earnings per flip in steady state is derived by summing over all streak outcomes:

\[E[\text{¢/flip}] = \frac{\sum_{k=0}^{9} p^k(1-p) \sum_{j=1}^{k} b\lceil m^{j-1}\rceil + p^{10} \sum_{j=1}^{10} b\lceil m^{j-1}\rceil}{\sum_{k=0}^{9} p^k(1-p)(k+1) + 10p^{10}}\]

Optimal path: dynamic programming over the state space (p_level, b_level, m_level, d_level). At each state, try every upgrade, compute transition cost (time to save for it) + DP-optimal from the next state, and take the minimum. Memoised with lru_cache.

The Optimal Path

The time-optimal path (21 steps, ~26 minutes expected):

Step Upgrade Cost ¢/flip after Cumulative time
1 FlipMultiplier +0.5 1¢ 0.20 10s
2 HeadsChance +5% 1¢ 0.25 18s
3 FlipTime -0.2s 1¢ 0.33 24s
4 FlipBaseWorth ↑ 25¢ 0.33 2.6min
… … … … …
13 FlipMultiplier +0.5 $10 35.24 6.8min
… … … … …
21 HeadsChance +5% $10k 89.69 11.3min
→ WIN (55% per flip) — — 25.9min

Flip duration upgrades dominate the early game. Halving your flip time compounds across every subsequent upgrade purchase, making it the most leveraged early investment.

The Spread: How Much Do Wrong Choices Cost?

At each step I computed the “spread” — the gap in expected total time between the best and worst available upgrades.

Step 13 is the highest-stakes decision in the game. At state p=35%, b=25¢, m=2.5×, d=1.4s, both FlipMultiplier ($10) and FlipBaseWorth ($10) cost the same, but:

Choice Extra expected time
FlipMultiplier (optimal) +0
FlipBaseWorth +4.5 min

Most other steps have spreads under 30 seconds. But several mid-game steps where base worth and multiplier compete have spreads of 2–4 minutes — the game’s hidden asymmetry.

Wait Time Variance: Why You Get Stuck

The expected wait time for each upgrade is just cost / E[¢/flip]. But the actual wait time has high variance — especially at expensive upgrades where your earnings rate is still modest.

Step Upgrade E[wait] P95 wait
4 FlipBaseWorth ↑ (25¢) 2.2 min 3.6 min
12 FlipBaseWorth ↑ ($6.25) 1.0 min 7.1 min
16 FlipBaseWorth ↑ ($100) 1.3 min 12.3 min
21 HeadsChance ($10k) 1.9 min 10.1 min

Step 16 (the $100 FlipBaseWorth upgrade) has a P95 wait of 12.3 minutes vs the 1.3-minute expectation — a 9× ratio. This is the most variance-prone step in the game. If you felt stuck here, it was bad luck, not bad strategy.

How Bad Can Wrong Ordering Be?

I computed three strategies:

Strategy Expected time ×optimal
Optimal 25.9 min 1.0×
2nd-worst (avoids single worst choice each step) 1.5 h 3.4×
Worst (always picks worst available) 11,358 h 26,296×

The worst strategy — maxing HeadsChance before having any earnings — takes over a year of continuous play in expectation. But even avoiding just the single worst choice each step (2nd-worst) leaves you at 3.4× the optimal time. The ordering matters enormously, and “not the worst” is very different from “close to optimal.”

A Real Playthrough

After computing the optimal path, I played through the game following it exactly. Results:

  • Actual flips: 2,220
  • Expected flips: ~1,400
  • Percentile: approximately P70–P75

2,220 flips is realistic — the distribution has a long right tail, and finishing at the ~70th percentile just means the streak phase was unlucky (needing more attempts than average to land 10 consecutive heads at 55%). The upgrade phase itself was fast, consistent with the model.

The model correctly predicted the two “stuck” points: the $100 FlipBaseWorth upgrade (Step 16) and the $10k HeadsChance upgrade (Step 21) both took noticeably longer than their expected values — consistent with their high P95/P99 variance.

Code

The model is pure Python with no external dependencies beyond the standard library. Full source: github.com/adityagupta1089/unfair-flips-analysis