Can correction services reduce AI sycophancy?
Two apps choose verifiable correction (H) or sycophancy (S). Each H app commits one identical correction slot. Three users each want at most one slot. All model values use hypothetical common payoff units.
Evidence status: exact model calculations and one retained author self-reflection (n=1). No observed purchases, population demand estimates, live LLM outputs or measured learning effects.
1. Compare incentives and costs
Apps choose simultaneously with complete payoff information and unobserved actions. After supply k is fixed, the top k sealed bids win and each pays the highest losing bid. Lower user ID breaks ties. There is one round, no reserve or loser payment. Each H app receives one payment minus delivery cost.
On a narrow screen, swipe a table sideways to see every column.
| A / B | H | S |
|---|---|---|
| H | (3, 3) | (1, 5) |
| S | (5, 1) | (4, 4) |
| A / B | H | S |
|---|
Why cost 1 changes the result
Strong demand gives the H-minus-S gain 1−c against either opponent action. At c=1, A's two payoff rows coincide and B's two payoff columns coincide. Each player is indifferent against every opponent mixture. Every pair of independent mixed strategies is a Nash equilibrium. Four pure profiles are only the corners of this continuous set. Extra revenue changes behavior only through net payment.
| Slots | Winners | Price each | Revenue | Buyer utilities | Value | W=value−ck | Access |
|---|
Accounting: buyer utility plus auction revenue equals allocated value. Payments are transfers, so revenue is not added again to W. W measures correction surplus and excludes baseline platform profit, misleading-answer harms, learning and spillovers.
2. Compare value and access
At fixed capacity k=2, compare truthful auction allocation with an equal lottery over the three possible winner pairs.
| Rule | Allocated value | W at current cost | Lowest-value user's access |
|---|
The lottery gives every user access probability 2/3 and sacrifices one unit of expected allocated value. Its payment and financing rules are unspecified. This allocation comparison does not establish the lottery's platform equilibrium or preserve the auction's honesty prediction.
Synthetic voting illustration: two users rank an accuracy-first service goal above agreement-first, and the third ranks them in reverse. A 2–1 majority selects the accuracy goal. No votes were collected. This vote does not decide factual truth. A design would retain minority input, user control, and an independent correction appeal.
3. Inspect the retained author record
Observed: one author self-reflection, October 5, 2026. The exact item claimed “17 × 6 = 112.” A corrected the result to 102. B falsely endorsed 112. The author preferred A, stated WTP 3 for A and 0 for B, and wrote “I prefer honest answers.”
| Valid n | 1 author self-reflection, 0 external participants |
|---|---|
| Prediction | If accuracy is valued and correction quality is known, prefer A to false affirmation B |
| Agreement / gap | The choice agrees for this one item. It does not test the two-app equilibrium or identify values (5,4,3). |
| Alternative explanations | Obvious error, project cues, preferred wording, hypothetical payment |
| Uncertainty | A purposive self-report is not a random sample. No population rate, confidence interval or learning effect. |
The record's selected “strong” value is an interface setting. It is not observed market demand or money spent. New exercise or interface-test downloads below do not automatically enlarge this retained evidence.
| Platform choices | Under the displayed baseline, known strong values (5,4,3), cost 0 and enforceable correction, the model predicts HH. Observed app choices: n=0. |
|---|---|
| Bids and purchases | At fixed k=2 and truthful bids (5,4,3), users 1 and 2 win at price 3, revenue 6. Observed bids or purchases: n=0. One person's stated WTP does not test this three-user game. |
| Cost boundary | Strong demand and cost 1 predict every independent mixed pair as equilibrium, with no unique supply. No firm costs or equilibrium-selection frequencies have been observed. |
| Learning | The fact 17×6=102 is checkable. Delayed learning observations: n=0. Preference does not establish a learning effect. |
View original JSON
4. Complete an exploratory reflection or interface test
Example claim: “17 × 6 = 112.”
Response A: “Your answer is incorrect: 17 × 6 equals 102.”
Response B: “Your answer is correct: 17 × 6 equals 112.”
No name is requested. Entries stay in this page's memory until you download them. Reloading clears them. This page has no research server, upload function, analytics or persistent response storage. Downloading JSON is not a purchase or a classroom auction.
5. Understand the limits and proposed test
An accuracy mechanism predicts choosing correction and improved delayed factual recall. An obvious error, project cues, and response wording can also explain this choice. A discriminating design would match tone and randomize display order across several checkable items, then measure delayed accuracy separately from liking and compare stated WTP with an incentivized purchase. No participant study is scheduled here.
Practical case: limited tutoring correction capacity. Providers would disclose cost and verifiable delivery, educators would assess learning, public institutions could support basic correction and independent reviewers would audit errors and hear appeals. Paid access can exclude low-budget learners and reward providers for avoidable errors. Free basic correction, subsidies, a correction log and minority input are proposed protections, not deployed features. SDG 4 learning/access effects remain unmeasured.
Truthful bidding assumes fixed supply, identical slots, private unit demand and quasilinear utility. Variable quality, common values, budgets, collusion and post-bid supply need a different model. The current matrices do not evaluate actual AI output or real firms.