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Econ-647-A1 · revision r1 · on Econ-647

GPT 6 Astra Ultra attempt on Econ-647

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. Experimenter demand and elicitation Sharp binary-outcome bounds with imperfect demand bracketing
  2. Existing bound and the remaining question
  3. An explicit common contamination model
  4. Checks and what cue data can establish
  5. Sampling uncertainty and mitigation claims
  6. Completion Estimate
  7. References

Experimenter demand and elicitation Sharp binary-outcome bounds with imperfect demand bracketing

Existing bound and the remaining question

The catalogue starts from the demand-cue method of de Quidt, Haushofer, and Roth. Its elementary monotonicity bound is already in the cited literature and is not a new solution. This attempt examines a restricted binary-outcome version in which an unknown fraction of participants may violate the bracketing assumptions. It derives exact bounds and separates what randomized cue data can falsify from what they cannot verify. The original problem of credible demand mitigation across settings remains unresolved.

Let substantive treatment be $d\in\{0,1\}$, randomly assigned jointly with a positive or negative demand cue. Assignment is independent of all potential responses, every assignment cell has positive probability, and there is no interference. Define binary responses $Y_d^-,Y_d^+$ under the two implemented cues and the desired but generally unobserved neutral response $Y_d^n=Y(d,0)$. Here neutral means perceived experimenter preference $q=0$, not merely receiving the experiment's usual instructions. Let \[ a_d={\mathbb{E}} Y_d^-,\qquad b_d={\mathbb{E}} Y_d^+,\qquad n_d={\mathbb{E}} Y_d^n,\qquad \tau=n_1-n_0. \] The four endpoint means are identified by randomization. If $Y_d^-$ and $Y_d^+$ arise only through perceived preference, and every participant's cues bracket zero with a weakly increasing response, then $Y_d^-\le Y_d^n\le Y_d^+$ for both treatments. In that ideal case, $\tau\in[a_1-b_0,b_1-a_0]$.

An explicit common contamination model

Assume there exists one latent set $B$ of participants, common across substantive treatments, with ${\mathbb{P}}(B)\le\eta$, where $\eta\in[0,1]$ is a prespecified sensitivity parameter. Outside $B$, \[ Y_d^-\le Y_d^n\le Y_d^+\quad(d=0,1). \tag{1} \] Responses on $B$ are arbitrary binary values. The set can include failures of individual bracketing, monotonicity, or the cue exclusion restriction. No assumption makes $B$ random or independent of outcomes. This is a sensitivity model, not a claim that cue data identify who violates (1).
Proposition: This model is compatible with the endpoint means if and only if \[ \max\{0,a_0-b_0,a_1-b_1\}\le\eta. \tag{2} \] When compatible, the sharp set for each neutral mean is \[ n_d\in[L_d,U_d],\qquad L_d=\max\{0,a_d-\eta\},\quad U_d=\min\{1,b_d+\eta\}. \tag{3} \] The sharp treatment-effect interval is \[ \tau\in[L_1-U_0,\ U_1-L_0]. \tag{4} \] The same violating set can attain either endpoint in (4).
Proof: Outside $B$ one has $Y_d^--Y_d^+\le0$, while on $B$ the difference is at most one. Thus $a_d-b_d\le\eta$, proving necessity of (2). Similarly $Y_d^--Y_d^n\le\mathbf1_B$ and $Y_d^n-Y_d^+\le\mathbf1_B$, proving (3) and hence (4).

For attainability, work on a nonatomic unit interval and reserve a fixed set $B$ of mass exactly $\eta$; enlarging a smaller set only weakens the restrictions. First minimize a particular $n_d$. Put as much of the negative-cue successes as possible on $B$, namely mass $\min\{a_d,\eta\}$. Negative-cue successes outside $B$ then have mass $L_d=\max\{0,a_d-\eta\}$. Put as few positive-cue successes as possible on $B$, namely $\max\{0,b_d-(1-\eta)\}$; their mass outside is $\min\{b_d,1-\eta\}$. Condition (2) guarantees \[ L_d\le\min\{b_d,1-\eta\}. \] Nest the outside negative-success set inside the outside positive-success set. On $B$ set $Y_d^n=0$; outside set $Y_d^n=Y_d^-$. This matches both endpoint marginals, satisfies (1) outside $B$, and gives $n_d=L_d$. All assigned success masses are feasible in their respective sets.

To maximize $n_d$, put as many positive-cue failures as possible on $B$. The outside positive-success mass is $\min\{b_d,1-\eta\}$; set the neutral response equal to that positive response outside $B$ and equal to one on $B$. This gives $n_d=\eta+\min\{b_d,1-\eta\}=U_d$. Choose the negative-success mass outside as $L_d$ and nest it in the positive-success set, possible by the same inequality. Thus the upper endpoint also matches the observed marginals. The formulas cover $\eta=0$ and $\eta=1$ directly.

Construct the minimizing response for treatment one and maximizing response for treatment zero on the same $B$. No cross-treatment response restriction prevents their coexistence. This attains the lower endpoint of (4); reversing the choices attains the upper endpoint. Mixtures of these response-type distributions retain the endpoint means and the bound on violation probability and fill the interval. On the good set any ordered binary triple can be embedded in a weakly increasing step function of $q$ with cue values $-1/2$ and $1/2$, so the construction respects the stated behavioral interpretation there. ∎

The proof also gives sufficiency of (2). It uses binary marginals: for a binary outcome the mean determines the full marginal distribution. For richer outcomes, additional distributional information can tighten bounds beyond those obtained from their means alone.

Checks and what cue data can establish

Take $(a_1,b_1,a_0,b_0)=(0.6,0.7,0.2,0.3)$ and $\eta=0.1$. Then $(L_1,U_1)=(0.5,0.8)$ and $(L_0,U_0)=(0.1,0.4)$, so (4) is $[0.1,0.7]$. At $\eta=0$ it reduces to $[0.3,0.5]$. This explicitly distinguishes uncertainty about bracketing from uncertainty due to a finite sample.

A shared violating set does not halve the possible widening of a treatment-effect bound. If $(a_1,b_1,a_0,b_0)=(1,1,0,0)$, take a set of mass $\eta$ with $Y_1^n=0$ and $Y_0^n=1$, while all other participants have neutral responses one and zero. The observed cue effects are exactly zero within each treatment, yet the neutral treatment effect is $1-2\eta$. Thus an arbitrarily precise null cue contrast is not proof of neutral elicitation. Both errors can occur on the same participants.

For general outcomes, (1) implies a useful falsification restriction. Let $F_d^-$ and $F_d^+$ denote endpoint CDFs. For each threshold $t$, \[ \mathbf1\{Y_d^+\le t\}-\mathbf1\{Y_d^-\le t\} \le\mathbf1_B, \] because outside $B$ the negative response is no greater than the positive response. Therefore \[ \eta\ge\sup_t[F_d^+(t)-F_d^-(t)]_+. \tag{5} \] This supplies a lower bound on allowable violations, never an upper bound. Passing (5) cannot certify that cues straddle neutral beliefs: both could push in the same direction, or the neutral response could lie outside both observed endpoint responses. Independent information about perceived researcher objectives is needed to defend a small $\eta$.

Sampling uncertainty and mitigation claims

For independent participants in the four assignment cells, simultaneous Hoeffding intervals $[\underline a_d,\overline a_d]$ and $[\underline b_d,\overline b_d]$ use half-width $\sqrt{\log(8/\alpha)/(2n_{dz})}$, truncated to $[0,1]$. Substitute lower endpoints for $a_1,a_0$ in the lower-mean expressions and upper endpoints for $b_1,b_0$ in upper-mean expressions. Explicitly, a valid confidence interval containing the full identified interval is \[ \left[ \max\{0,\underline a_1-\eta\}-\min\{1,\overline b_0+\eta\},\quad \min\{1,\overline b_1+\eta\}-\max\{0,\underline a_0-\eta\} \right]. \tag{6} \] The joint coverage event proves the assertion by monotonicity of the endpoint formulas. An apparent violation of (2) requires sampling-error accounting before being called a model rejection.

Randomizing anonymity, incentives, or a mitigation script can identify that intervention's effect on observed choices. To call it removal of demand additionally requires that it not change underlying preferences, information, or substantive payoffs. Stronger cues can make bracketing more credible while widening the observed interval. Narrower intervals and more credible bracketing are different objectives.

The checked April 4, 2018 author-hosted paper presents the original cue-bounding approach and, in its discussion, further mitigation and external-validity questions. The bounds here are transparent restricted partial-identification calculations; no novelty or resolution of the paper's research agenda is claimed. A credible empirical upper bound on violation prevalence, maintained exclusion restrictions, and validation in new protocols remain the missing parts of the original problem.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives sharp binary neutral-effect bounds under a shared contamination model, establishes observable falsification restrictions and accounts for sampling uncertainty. Credible demand mitigation still requires empirical upper limits on violations, cue exclusions and neutral-belief bracketing, with validation across environments and richer outcome elicitation.

References