Exploration Reward Model

• Exploration Reward Model is a formalism that designs calibrated incentives to encourage informative, non-myopic exploration in reinforcement learning and multi-armed bandit settings.
• It employs a time-expanded signaling policy that randomizes between myopic selection and payment-based exploration, optimized via convex programming for worst-case performance.
• The model integrates agent heterogeneity and budget constraints to align immediate rewards with long-term, high-reward decision-making.

An exploration reward model is a formalism that defines how a principal (system designer or platform) incentivizes a sequence of myopic agents to perform informative, non-greedy actions in an uncertain environment, typically in the multi-armed bandit or reinforcement learning setting. Exploration rewards are critical in settings where the agent’s or user’s natural incentives are insufficient to ensure optimal long-term outcomes, especially when immediate rewards encourage suboptimal exploitation. Such models precisely characterize the algebraic, statistical, and algorithmic mechanisms by which additional incentives—typically monetary or otherwise—are calibrated to induce sufficient exploration for information gathering, enabling asymptotically optimal, high-reward policies.

1. Formal Problem Structure and Motivation

The canonical exploration reward model is instantiated in sequential decision making where a principal wishes to maximize a discounted sum of expected rewards:

$R = \sum_{t=0}^\infty \gamma^t \mathbb{E}[v_{i_t}]$

where $v_{i_t}$ is the reward of the arm $i_t$ chosen at time $t$, and $\gamma$ is the discount factor. Agents, however, are myopic and maximize only the expectation of the immediate reward. This creates a misalignment: without further incentives, agents select arms greedily, never exploring potentially higher-reward options whose value is uncertain, preventing long-run optimal learning and exploitation.

To realign agent and principal incentives, the principal can offer per-action monetary bonuses $c_i$, modifying each agent’s effective utility to:

$\mathbb{E}[v_i] + \mu(c_i)$

where $\mu$ is the agent’s utility function for money. The central problem is then to design a sequence of (possibly randomized, signal-dependent) payments to maximize the principal’s long-term reward while satisfying budget, fairness, or robustness constraints.

2. Heterogeneity and Signaling in Agent Money Utility

Real-world agents possess heterogeneity in their value for money, quantified via a non-linear and agent-specific function $\mu$. The principal has partial, noisy information about each agent’s conversion ratio $r$, which represents the marginal utility of money for that agent. This information is formalized as a signal $s$ realized upon the agent’s arrival, yielding a conditional distribution $F_s$ over $r$. Worst-case analysis (in terms of regret or cost) shows that the non-linear $\mu$ can be reduced to the linear form $\mu(x) = r x$, since linear functions dominate non-linear ones in the relevant convex ordering for these problems.

The model thus becomes signal-dependent: each agent’s observed signal updates the principal’s posterior $F_s$ regarding the agent’s conversion ratio, and payment decisions are tailored accordingly.

3. Time-Expanded and Signal-Dependent Policies

The core policy architecture, termed time-expanded signaling (TES), randomizes between two regimes for each agent (given their signal $s$):

• With probability $q_s$, allow myopic selection (no payment).
• With probability $1 - q_s$, offer a payment just sufficient to induce selection of the exploratory (non-myopic, e.g., Gittins index) arm.

The required payment for the exploratory action $i_t^*$, conditional on signal $s$, is:

$c_{t,i_t^*} = \frac{x - y}{F_s^{-1}(q_s)}$

where $x = \max_i \mathbb{E}[v_i]$ and $y = \mathbb{E}[v_{i_t^*}]$. This payment is calibrated so that only agents with the top $(1-q_s)$-quantile of conversion ratios (as inferred from their signal) are incentivized to explore against their myopic preference.

4. Convex Program Formulation and Approximation Guarantees

The principal computes the full vector of myopic probabilities $q = (q_s)$ by solving a convex program that optimally trades off expected exploration cost and reward: $$\begin{align*} \text{minimize} & \quad \sum_s \pi_s q_s\ \text{subject to} & \quad \sum_s \pi_s q_s \geq \lambda \sum_s \pi_s \frac{1-q_s}{F_s^{-1}(q_s)}\ & \quad 0 \leq q_s \leq 1\ \forall s \end{align*}$$ Here, $\pi_s$ denotes the prior probability of signal $s$, and $\lambda$ is a Lagrange multiplier controlling the cost-reward tradeoff. The solution value $p^* = \sum_s \pi_s q_s$ sets the worst-case performance ratio: TES($q^*$) achieves at least a fraction $(1 - p^* \gamma)$ of the optimal value, regardless of specific bandit instance.

This convex program is computable under model knowledge of signal distributions and agent-type priors.

5. Information Monotonicity and Signal Value

A rigorous monotonicity result is established for the value of information provided by the signaling structure. If one signaling scheme is a garbling (in the sense of Marschak) of another, the approximation ratio of the optimal exploration policy does not improve:

$1 - p^*(\phi)\gamma \geq 1 - p^*(\phi')\gamma$

where $p^*(\phi)$ and $p^*(\phi')$ correspond to the optimal myopic probabilities under the original and garbled signals, respectively. This strictly quantifies the benefit of finer agent information for the principal.

6. Budgeted Exploration and Extension to Payment Constraints

The framework readily accommodates strict payment constraints. Specifically, when the principal’s total expected expenditure is capped at $b \cdot OPT_\gamma$, there exists a randomized mixture of TES policies achieving:

$$\min_\lambda \{~1 - p^*(\lambda)\gamma + \lambda b~\}$$

where policy mixing and convex optimization guarantee budget feasibility while preserving robust worst-case reward guarantees, as each policy’s $q$-vector can be tuned and mixed to exactly exhaust the allowed budget.

7. Worst-Case Instances and Tightness: Diamonds in the Rough

To demonstrate the tightness of the approximation guarantees, the paper constructs "Diamonds in the Rough" instances—multi-armed bandit settings with one safe arm (constant moderate reward) and infinitely many risky arms (rare, extremely high reward, otherwise zero). These instances are proven to match the theoretical lower bound: no policy can outperform the TES guarantee of $(1 - p^*\gamma)$. By applying Karush–Kuhn–Tucker (KKT) analysis, the paper shows the convex program’s solution coincides with that for the worst-case instance, validating the optimality of TES in the hardest environments.

Table: Structural Summary of the Exploration Reward Model

Component	Description	Mathematical Object / Condition
Agent utility	Heterogeneous, non-linear $\mu$	$\mathbb{E}[v_i] + \mu(c_i)$
Signal	Partial information on agent value for money	Updating prior $F \to F_s$
Payment Calculation	Signal-dependent, threshold-based	$c_t = (x - y)/F_s^{-1}(q_s)$
Policy optimization	Robust, worst-case optimal via convex program	Min $\sum_s \pi_s q_s$, s.t. feasibility constraint
Monotonicity	Value of finer signals non-decreasing	$1 - p^*(\phi)\gamma \geq 1 - p^*(\phi')\gamma$
Budget constraint	Policy mixture to exhaust payment budget	$$\min_\lambda \{1 - p^*(\lambda)\gamma + \lambda b\}$$
Tightness instance	"Diamonds in the Rough" constructions	KKT conditions establish bound is tight

8. Implications and Applications

This exploration reward model applies broadly to principal–agent incentives in platforms, online experimentation, and algorithmic recommendation systems where agents are non-strategic (myopic) but the principal requires long-run optimal decisions. The convex programming approach yields robust, efficiently computable policies even when agents’ responsiveness to incentives varies arbitrarily and is only partially observable. The signal monotonicity property quantifies organizational value of agent information systems, and the theoretical tightness via worst-case instances establishes practical limits of incentive-based exploration under heterogeneity.

The model provides a unified, worst-case-optimal prescription for exploration incentives in crowdsourcing, online marketplaces, and other settings where exploration dynamics are governed by strategic incentive alignment between a central organizer and a heterogeneous population of agents.