Research map / Probabilistic and scientific modeling / Treatment effects
Causal Inference from Competing Treatments
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Builds a game-theoretic model in which advertisers compete for subject slots and ranks to estimate a causal effect, showing that a tractable sample-value objective approximates the estimation-error game and has pure Nash equilibria.
Problem
Causal effect estimation from RCTs when multiple treatment administrators compete for subjects' attention, so treatments are shown at different positions with position-dependent effectiveness; how should budgets be allocated strategically to minimize estimation error?
Approach
Model competing administrators bidding on subject slots under a probabilistic rank allocation; show the minimax estimation error is governed by a sample value sum_r n_r alpha_r^2, and that Nash equilibria of the tractable sample-value game are approximate equilibria of the estimation-error game when budgets are large (B = omega(k^2)).
Key results
- Theorem: the sample-value game G_SV has a pure Nash equilibrium for any discount sequence with alpha_1=1 and alpha_r in (0,1); equal uniform splitting with maximal competition avoidance is always an equilibrium when B <= n.
- Any Nash equilibrium of G_SV is an (epsilon, eta)-approximate Nash equilibrium of the estimation-error game with epsilon = O(k/sqrt(B)) and eta = O(exp(-B/k^2)), so with B = omega(k^2) the approximation is (o(1), o(1)).
- Minimax lower bound on MSE is min(sigma^2/(16(1-q) sum_r n_r alpha_r^2), 1); the pure equilibrium conjecture is proven only for k=2 administrators.
Surprising findings
- Estimating discount factors jointly from pilot data via data splitting does not reduce estimation error compared with using only first-rank data.
Limitations
- Conjecture that the only pure Nash equilibria are the uniform-split allocations is proven only for two administrators
- Analysis assumes known discount factors alpha_r and a common relevance probability p across admins
- Approximation guarantees require large budgets relative to k^2 and bounded per-slot bids
Experimental setup
Metrics: estimation error (MSE) approximate Nash equilibrium (epsilon, eta)
Builds on: Le Cam's method McDiarmid's inequality inverse variance weighting Horvitz-Thompson estimator
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