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Delegate Pricing to Algorithms: When Slow and Steady Wins the Race

Galit Ashkenazi-Golan, Edward Plumb, Clemens Possnig, Yufei Zhang

arXiv:2610.11675v1econ.THcs.GT

TL;DR

Prior work largely treats learning rules as fixed, leaving the strategic choice of learning speed and profits during adaptation less studied. The paper models firms choosing gradient learning rates in a continuous-time pricing game and finds that incentives differ by stage-game strategic interaction. Complementarity can support deliberately moderate learning despite eventual convergence to competitive prices.

  • Problem

    Prior work largely treats learning rules as exogenous and emphasizes asymptotic outcomes, leaving strategic learning-rate choice and profits along adaptation understudied.

    Source: §1 · pp. 1–2

  • Method

    The paper models a continuous-time Bertrand Prisoner’s Dilemma in which firms choose learning rates for gradient dynamics and evaluate the strategic learning-rate meta-game.

    Source: §1 · p. 2, §2 · p. 5

  • Results

    With strategic substitutability, both firms find acceleration profitable and no equilibrium exists; with complementarity, open sets of payoffs and initial conditions support interior equilibria with moderated learning rates.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    Firms can extract rents by slowing convergence even when algorithms ultimately reach a competitive equilibrium, so analysis should examine ex ante choices governing adjustment speed.

    Source: §4 · p. 13

    The model assumes a Prisoner’s Dilemma payoff ordering, T > R > P > S, with low prices the unique dominant-strategy equilibrium.

    Source: §2 · p. 5

Abstract

from arXiv · show

We study the strategic design of learning algorithms in a canonical continuous-time pricing game. We introduce a meta-game in which firms select learning rates for a gradient dynamic, evaluating payoffs as the discounted sum of profits accrued along the entire learning path. We uncover a fundamental dichotomy driven by stage-game incentives: if the stage game exhibits strategic substitutability, firms unambiguously prefer the fastest possible algorithms to rapidly exploit a gradually adjusting opponent. Under strategic complementarity, however, an excessively fast algorithm accelerates the rival's competitive response, destroying transitional profit margins. Even though the competitive price is a strictly dominant action in the underlying stage game, we prove that firms optimally design sluggish algorithms, extracting surplus during a prolonged convergence to the competitive equilibrium.

1 Introduction

This introduction reframes algorithmic pricing as a strategic choice over learning speed: firms care about profits during adjustment, and the resulting incentives depend on the stage-game interaction. It positions the framework against work on algorithm choice, commitment, prescribed learning, and adaptive agents.

1 Introduction

  • The paper addresses the understudied choice of learning path by modeling firms that choose gradient learning rates and value profits earned during adaptation.The literature often treats learning rules as exogenous, while firms can tune hyperparameters and optimize profits throughout learning.Source: §1 · p. 2
  • The learning-rate meta-game’s equilibrium predictions depend on whether relative undercutting gains make the stage game strategically substitutable or complementary.The baseline Prisoner’s Dilemma converges to the competitive Nash equilibrium regardless of learning rates, isolating the role of profits along the adjustment path.Source: §1 · p. 2
  • Under strategic substitutability, acceleration slows the opponent’s convergence, making further acceleration profitable for both firms and eliminating a learning-rate equilibrium.This mechanism lets a firm exploit its opponent for longer during adjustment.Source: §1 · pp. 2–3
  • Under strategic complementarity, the rival’s accelerated response can outweigh the direct benefit of faster adjustment, supporting interior equilibria with moderate learning rates.The paper characterizes open sets of payoffs and initial conditions for these equilibria.Source: §1 · p. 3
  • The response to weaker complementarity depends on the normalized loss from a rival’s switch to low pricing: learning rates rise above a threshold and fall below it.Above the threshold, slower learning avoids inducing rival acceleration; below it, slow learning prolongs the high-price transition.Source: §1 · p. 3

1.1 Related Literature

  • The framework offers analytical results for a hyperparameter meta-game, distinguishes transition-path rents from altered terminal outcomes, and differs from numerical algorithm-choice work [7] [16] and opponent-aware updates [10].Unlike, firms commit to a parameter of a fixed learning rule rather than a price-response function; prescribed learning studies [6, 3] instead ask whether a noncompetitive long-run outcome is sustained.Source: §1.1 · pp. 4–5

2 The Model

The paper models firms competing over high and competitive prices while delegating price adjustment to autonomous gradient-learning agents. Firms choose learning rates to maximize discounted profits over the adjustment path, not just the eventual equilibrium.

2 The Model

  • Firms choose between a high supra-competitive price and a low competitive price in a repeated Prisoner’s Dilemma, whose unique Nash equilibrium is the competitive price pair.The market state records each firm’s probability of choosing the high price.Source: §2 · p. 5

2.1 Algorithmic Delegation and Learning Dynamics

  • Autonomous agents adjust pricing strategies using projected gradient dynamics, while firms choose learning rates that control the speed of adjustment.Projection keeps the state within the feasible probability space.Source: §2.1 · pp. 5–6

2.2 The Meta-Game

  • Although all learning-rate profiles lead to the competitive equilibrium, firms earn profits throughout the adjustment path, making that path strategically relevant.Each firm maximizes discounted accumulated profit from a deterministic trajectory given the initial market condition.Source: §2.1 · p. 6, §2.2 · p. 6

2.3 Example

  • In the numerical example, accelerating one firm’s move to the competitive price can speed its rival’s adjustment and reduce the first firm’s payoff.With the rival’s rate fixed at 1 and initial probabilities at 0.9, Firm 1’s discounted utility is maximized at an interior learning rate.Source: §2.3 · p. 7, Figure 1 · p. 8

3 Pricing, Fast and Slow

The sign of the stage game’s strategic interaction determines whether firms gain from accelerating or moderating their learning rates. Under complementarity, an interior learning-rate equilibrium can arise because a firm’s acceleration hastens its rival’s adjustment and reduces profits during the transition.

3 Pricing, Fast and Slow

  • The sign of δ distinguishes the regimes: acceleration slows a rival under strategic substitutes but increases the rival’s incentive to undercut under strategic complements.In the complements case, capacity constraints, tipping points, or strong brand loyalty are cited as characteristic market features.Source: §3 · p. 8

3.1 Strategic Substitutes

  • Under strategic substitutes, a firm’s faster adjustment slows its rival’s convergence, so each firm’s payoff strictly increases with its own learning rate and no finite-rate Nash equilibrium exists.If a maximum rate is available, both firms prefer it.Source: §3.1 · pp. 9–10

3.2 Strategic Complements

  • A firm’s best response under complementarity can be non-monotone because moving too quickly to the competitive price shortens the period in which it can exploit the rival’s still-high prices.The equilibrium learning rate scales with the discount rate, so more patient firms learn more slowly.Source: §3.2 · pp. 10–11
  • Under strategic complementarity, an open set of initial conditions and payoffs supports a finite, positive symmetric equilibrium learning rate proportional to the discount rate.The equilibrium is a strict global symmetric Nash equilibrium, and the theorem characterizes conditions using the normalized temptation spread.Source: §3.2 · pp. 10–11
  • The equilibrium response to weaker complementarity depends on the normalized loss from a rival switching to low pricing: above the threshold the rate rises, while below it the rate falls.Above the threshold, moderation chiefly avoids inducing faster rival adjustment; below it, moderation helps prolong the high-price transition.Source: §3.2 · pp. 12–13

4 Conclusion

The conclusion argues that strategic learning-rate choices can prolong elevated prices even when standard learning converges to a competitive equilibrium. It also identifies broader applications and an extension in which firms choose initial algorithm states.

4 Conclusion

  • Firms can extract rents during convergence by tuning learning rates to delay convergence, despite eventual convergence to the competitive Nash equilibrium.The conclusion argues that assessing algorithmic pricing solely at steady state is insufficient.Source: §4 · p. 13
  • The framework also yields an algorithmic arms race in zero-sum settings and allows asymmetric learning rates to select equilibria in coordination games.Source: §4 · p. 13
  • Future work could endogenize algorithm initialization, where choosing initial states alongside learning rates might make initialization near mutual cooperation a robust equilibrium.Source: §4 · p. 13

A.1.1 Proof of Proposition 3.1

The proof establishes the trajectory’s sensitivity to a learning-rate change by first deriving local derivative signs, then extending them over the full interval and using symmetry.

A.1.1 Proof of Proposition 3.1

  • The proof uses parameter dependence of differential equations to define derivatives of the two trajectories with respect to Firm 1’s learning rate.Source: §A.1.1 · p. 15
  • For sufficiently small positive times, the proof establishes u(t) < 0 and v(t) > 0 from continuity and the differential relations.Source: §A.1.1 · p. 15
  • A contradiction argument extends u(t) < 0 over the interval, implies v(t) > 0, and symmetry supplies the corresponding result for the other player.Source: §A.1.1 · pp. 15–16

A.1.2 Proof of Proposition 3.2

The proof establishes that Firm 1’s payoff is strictly increasing in its learning rate by signing the payoff derivative before and after a player reaches zero.

A.1.2 Proof of Proposition 3.2

  • If the players reach zero at different times, the remainder term is positive after applying the post-boundary sensitivities and chain-rule cancellation.When Tx = Ty, the remainder is zero; the proof handles the two unequal boundary-order cases separately.Source: §A.1.2 · pp. 16–17
  • Before either player reaches zero, the derivative integrand is strictly positive because the (S − P)∂γ1xγ contribution is positive.The proof uses S < P, xγ(t) ≤ 1, and ∂γ1yγ ≥ 0; when δ = 0, the other contribution vanishes.Source: §A.1.2 · p. 16
  • ∂γ1J1 > 0 for every case, so Firm 1’s payoff is strictly increasing in γ1.The proof signs the derivative’s integrand and remainder term, accounting for which player reaches zero first.Source: §A.1.2 · pp. 16–17

A.1.3 Proof of Theorem 3.3

Because each firm’s payoff strictly increases with its own learning rate, no learning-rate pair is a Nash equilibrium and every best-response correspondence is empty.

A.1.3 Proof of Theorem 3.3

  • Firm 1’s payoff is strictly increasing in γ1 for any fixed γ2 > 0.Thus any candidate with positive learning rates admits a profitable increase in Firm 1’s rate.Source: §A.1.3 · p. 17
  • For any candidate rate pair, Firm 1 can raise γ1 by some ε > 0 and obtain a higher payoff.The proof takes γ1 ≤ γ2 without loss of generality.Source: §A.1.3 · p. 17
  • No Nash equilibrium exists, and BRi(γ−i) = ∅ for every positive opponent rate and either firm.Source: §A.1.3 · p. 17

A.1.4 Proof of Proposition 3.4

The proof of Proposition 3.4 follows the proof of Proposition 3.1, with sensitivity signs changing because δ changes sign.

A.1.4 Proof of Proposition 3.4

  • The proof of Proposition 3.4 is analogous to the proof of Proposition 3.1.Source: §A.1.4 · p. 17
  • The sensitivity signs change only because δ changes sign.Source: §A.1.4 · p. 17
  • The stated proof difference is limited to the sign change in sensitivities due to δ.Source: §A.1.4 · p. 17

A.2 Undiscounted Game

The appendix analyzes the undiscounted game by establishing scaling and payoff bounds, characterizing when the symmetric learning-rate ratio is optimal, and giving conditions for convergence and an interior root of E.

A.2 Undiscounted Game

  • Multiplying both learning rates by the same positive factor rescales the trajectory in time, and the undiscounted excess payoff has a corresponding homogeneity property.The trajectory under rates λγ is the original trajectory evaluated at time λτ.Source: §A.2 · pp. 18–19
  • When E(x0) = 0, the symmetric rate ratio ρ = 1 is the unique strict global maximizer of the undiscounted payoff, with negative second derivative.For a fixed initial condition, the payoff has at most one critical point; if one exists, it is the unique strict global maximizer.Source: §A.2 · pp. 23, 26
  • For δ < 0, E(x0) has at most one zero on (0, 1] and is strictly negative to the right of any such zero.Thus, if E has a zero in (0, 1), E(x0) < 0 for all larger initial conditions up to 1.Source: §A.2 · p. 27
  • If E(1) < 0, then E has a unique root x̄ ∈ (0, 1), since E(x0) > 0 for all sufficiently small x0 > 0.The positivity near zero follows from f(0) = f′(0) = 0 and f′′(0) = 2δ²(T−P)(P−S) > 0.Source: §A.2 · p. 27
  • For δ < 0, a game-parameter constant C0 bounds the relevant payoff tail uniformly for all x0 ∈ (0, 1] and ρ ≥ 1.The bound supports control of the large-ρ portion of the payoff comparison.Source: §A.2 · p. 28
  • As x0 approaches x̄, the payoff difference in Lemma A.8 converges to zero uniformly over ρ ≥ 1.The proof combines a large-ρ tail bound with joint continuity on compact ρ intervals.Source: §A.2 · p. 28

A.3 Moving to the Discounted Game

This section analyzes how discounting affects learning-rate choices and symmetric equilibria, reducing discounted payoffs to an effective discount rate and establishing supporting convergence and regularity results. It also characterizes how the transversality constant governs nearby finite symmetric critical points and shows that, for initial conditions just above x̄, a learning rate can be each firm’s unique strict global best response, yielding a strict symmetric Nash equilibrium for every r > 0.

A.3 Moving to the Discounted Game

  • For an appropriately chosen initial condition, ρ = 1 remains the strict global symmetric Nash equilibrium for every discount rate r > 0.The section identifies ρ = 1 as the unique strict global maximizer of the undiscounted excess payoff.Source: §A.3 · p. 29
  • For fixed symmetric x0 = y0 ∈ (0, 1], W converges uniformly to J0 as r → 0+ on compact intervals and on [1, ∞).Source: §A.3 · p. 29
  • The scaling identity reduces (γ1, γ2, r) to ρ = γ1/γ2 and effective discount ˜r = r/γ2, with V1 = P + ˜rW(ρ, ˜r; x0).Source: §A.3 · p. 29
  • For x0 = y0 ∈ (0, 1], the analysis establishes a lower-tail penalty for sufficiently small ρ and discount rate.Source: §A.3 · p. 32
  • Near ρ = 1, W is twice continuously differentiable in ρ, its first derivative at ρ = 1 is continuously differentiable in effective discount and x0, and its second derivative at zero discount matches J0.Source: §A.3 · pp. 33, 37
  • The function D(q̄) has exactly one root q̄c in (0, 1).This uniqueness underlies the transversality threshold used in the theorem.Source: §A.3 · pp. 39, 42
  • T > 0 exactly when (T−P)/(P−S) > 2.130075, equivalently when q̄ > 0.33486 and D(q̄) > 0.The sign of T equals the sign of D(q̄), which crosses from negative to positive at its unique root q̄c ≈ 0.33486.Source: §A.3 · p. 43
  • For δ < 0, if T > 0, every r > 0 admits a finite symmetric interior critical point for x0 just above x̄; if T < 0, this holds just below.The theorem specifies an ε-neighborhood on the corresponding side of x̄.Source: §A.3 · p. 44
  • Setting γ*(r) = η*(x₀)r makes the effective rate equal to r̃*(x₀), so the symmetric profile is a critical point of firm 1’s payoff.The construction chooses r̃*(x₀) > 0 on the interval just above x̄ so that the payoff derivative at the symmetric profile is zero.Source: §A.3 · p. 45
  • Strict concavity near ρ = 1 and localization of the global maximizer establish that this critical point is the unique strict global payoff maximum.The limiting payoff has a unique strict global maximum at ρ = 1; uniform convergence confines the discounted-game maximizer to a compact interval and then to its neighborhood.Source: §A.3 · p. 46
  • For x₀ ∈ (x̄, x̄ + ε), (γ*(r), γ*(r)) is a strict global symmetric Nash equilibrium for every r > 0.Each firm’s rate is the other firm’s unique strict global best response.Source: §A.3 · pp. 45–46

A.4 Proof of Proposition 3.6

The proof tracks the symmetric equilibrium branch as complementarity changes, using smoothness and the implicit function theorem. The equilibrium learning-rate response has a sign determined by transversality and a threshold ratio.

A.4 Proof of Proposition 3.6

  • Joint continuous differentiability of Φ in effective discounting, the initial condition, and δ supports the implicit-function comparative statics.At zero effective discounting, ∂δΦ is positive with the initial condition held fixed.Source: §A.4 · p. 47
  • The local equilibrium branch remains a symmetric equilibrium under sufficiently small changes in complementarity δ.The proof uses continuity of trajectories, hitting times, and tail-bound constants to preserve strict concavity and uniform payoff gaps.Source: §A.4 · p. 48
  • The equilibrium learning-rate derivative has the sign of T0: it is positive when (T−P)/(P−S) > t̄ and negative when the ratio is below t̄.The numerator is positive, while the denominator has the sign of T0.Source: §A.4 · p. 48
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