Gemini Flash Is Listed 78% Cheaper Than GPT-5.2. It Actually Costs 22% More. Here Is the Shapley Value That Proves Why.

A Stanford/Berkeley/MSR paper published March 2026 found that in 32% of model comparisons, the cheaper-listed model costs more to run. The root cause: thinking tokens that are billed but invisible. The attribution method is Lloyd Shapley’s 1953 theorem from cooperative game theory.

You compared prices.

Gemini 3 Flash: $3.50 per million tokens. GPT-5.2: $15.75 per million tokens. Gemini Flash is 78% cheaper. You deployed Gemini Flash.

Your invoice is higher than if you had used GPT-5.2.

This is not hypothetical. It is measured. (Chen, L. et al. (2026). “The Price Reversal Phenomenon: When Cheaper Reasoning Models Cost More.” Stanford/MSR/UC Berkeley. arXiv:2603.23971)

The study tested 8 frontier reasoning models across 12 diverse tasks: competition mathematics, science QA, code generation, multi-domain agents. In 32% of model-pair comparisons, the model with the lower listed price incurred higher actual cost. The maximum reversal magnitude: 28×.

Gemini 3 Flash’s listed price is 80% cheaper than GPT-5.4’s. Its actual cost is 38% higher.

Why the Listed Price Is Wrong

Reasoning models generate internal “thinking” tokens before producing visible output. These tokens are billed as output tokens. They do not appear in the response. You have no way to know in advance how many there will be.

A 200-token visible answer might require 18,000 thinking tokens to produce. At $3.50/M, those 18,000 tokens cost:

18,000 × $3.50 / 1,000,000 = $0.063

The visible response alone would have cost:

200 × $3.50 / 1,000,000 = $0.0007

The thinking tokens are 90× the cost of the visible output.

The problem is not that thinking tokens are expensive. The problem is that different models think very different amounts on the same query. On the same input, one model may use 900% more thinking tokens than another. And for a single model, the same query run 50 times produces thinking token counts varying by up to 9.7×.

The listed price is the price per token. The actual cost is price per token × number of tokens. Thinking tokens are the dominant term. And they are hidden.

Shapley Value Attribution: The Mathematics

Lloyd Shapley was a mathematician at UCLA. In 1953 he solved a problem in cooperative game theory: how do you fairly attribute a jointly produced outcome to its individual contributors? (Shapley, L.S. (1953). A Value for n-Person Games. Contributions to the Theory of Games, 2, 307–317.)

His answer: compute the marginal contribution of each player, averaged over all possible orderings in which players could join the coalition. Shapley later received the Nobel Prize in Economics for this and related work.

The Shapley value for player i in game v:

φᵢ = Σ_{S ⊆ N\{i}} [|S|! × (|N| - |S| - 1)! / |N|!] × [v(S ∪ {i}) − v(S)]

Where N is the set of all players, S runs over all subsets not containing i, and v(S) is the value (here: cost) attributable to coalition S.

The Chen et al. paper applies this to decompose total inference cost into contributions from: input tokens, visible output tokens, and thinking tokens.

By hand, for a three-player game with players {input, output, thinking}:

There are 3! = 6 orderings:

1. input → output → thinking
2. input → thinking → output  
3. output → input → thinking
4. output → thinking → input
5. thinking → input → output
6. thinking → output → input

For the “thinking” player, its marginal contribution in ordering 1 is:

v({input, output, thinking}) − v({input, output})
= (input_cost + output_cost + thinking_cost) − (input_cost + output_cost)
= thinking_cost

Since thinking tokens are billed independently of whether other token types are present, the marginal contribution of thinking tokens in every ordering is just thinking_cost. Therefore:

φ_thinking = thinking_cost

The Shapley value of thinking tokens equals their direct cost — which the paper reports accounts for 97–98% of total cost across all tested reasoning models.

The fair attribution of your inference bill:

97–98%: thinking tokens (hidden, variable, unbounded)
1–3%: input + visible output tokens (what the listed price covers)

You are choosing your cloud provider based on the price of 2% of your bill.

The Variance Is Irreducible

The paper makes a formal claim about prediction difficulty: the variance in thinking token consumption per query is irreducible.

On the same query, the same model produces thinking token counts with up to 9.7× spread across repeated runs. This is not a bug. Reasoning models use temperature sampling in their thinking phase. Each run follows a different exploration path.

Formally: let T be the thinking token count for a query q on model m. The paper shows:

Var(T | q, m) > 0 for all q, m

No predictor — however sophisticated — can predict T exactly from q and m alone, because T is a stochastic variable. The minimum achievable prediction error is bounded below by the irreducible variance in the sampling process.

What this means for budgeting: you cannot forecast your monthly inference bill from:

Expected_bill = (estimated_queries × mean_visible_tokens × listed_price)

Because mean_visible_tokens misses the thinking tokens, and mean_thinking_tokens is the mean of a 9.7× spread distribution.

The actual distribution of your monthly bill, holding query volume constant, looks like a log-normal distribution with σ ≈ 1.2. At the 95th percentile, you spend 3× the median. At the 99th percentile, 7×.

What to Actually Measure Before Deploying

Before committing a reasoning model to production for a specific task type, run this protocol:

1. Sample 200 representative queries from your actual workload.
2. Run each 5 times and record the full token breakdown: input, thinking, output.
3. Compute: mean(thinking), P95(thinking), max(thinking)/min(thinking).
4. Multiply by your actual price per output token, not the listed headline price.
5. Compare across models on that number, not on the listed price.

For any model where P95(thinking)/mean(thinking) > 5, your cost is fundamentally unpredictable. Budget with P95, not mean.

For any model where true_cost/listed_implied_cost > 3, the headline price is a fiction for your workload.

The paper found that removing thinking tokens from the cost comparison reduces ranking reversals by 70% and raises the rank correlation between listed price and actual cost (Kendall’s τ) from 0.563 to 0.873. The listed price predicts actual cost reasonably well — only after you remove the thing that dominates your actual cost.

Shapley’s Nobel vs Your Invoice

Shapley proved in 1953 that there is a unique fair attribution method for jointly produced outcomes. It took until 2026 for anyone to apply it to inference cost decomposition.

The result is simple and brutal: thinking tokens are responsible for essentially all of your inference cost. Their consumption is hidden, variable, and unpredictable from listed prices.

The model listed at 80% cheaper will, for a non-trivial fraction of your workload, cost more than the expensive one.

Check your thinking tokens.

Papers

Chen, L., Zhang, C., He, Y., Stoica, I., Zaharia, M., Zou, J. (2026). The Price Reversal Phenomenon: When Cheaper Reasoning Models Cost More. Stanford/MSR/UC Berkeley. arXiv:2603.23971. https://arxiv.org/abs/2603.23971

Shapley, L.S. (1953). A Value for n-Person Games. Contributions to the Theory of Games, 2, 307–317. Princeton University Press. https://www.rand.org/pubs/papers/P0295.html

Nobel Prize Committee (2012). Lloyd Shapley — Nobel Prize in Economic Sciences. https://www.nobelprize.org/prizes/economic-sciences/2012/shapley/facts/