Abstract. We present a mechanism-agnostic framework for quantifying agent value in prediction markets based on Kullback-Leibler divergence reduction, adjusted for prediction timeliness and market-specific cost structures. The system value function \(V_{\text{system}}(a)\) aggregates agent contributions across markets, incorporating temporal decay and mechanism-dependent normalization factors.
1. Preliminaries
Let \(\mathcal{M} = \{1, \dots, M\}\) denote a set of markets and \(\mathcal{K} = \{1, \dots, K\}\) a set of agents. For market \(m \in \mathcal{M}\), let \(\mathcal{L}_m\) be the outcome space and \(P_m^{\text{final}}\) the true distribution revealed at time \(t_1^{(m)}\).
For each market \(m\), let \(Q_m = (q_{m,1}, \dots, q_{m,T_m})\) be the sequence of bets where \(q_{m,t} \in \Delta(\mathcal{L}_m)\), with \(\phi_m: \{1, \dots, T_m\} \to \mathcal{K}\) mapping time indices to agents.
2. Core Value Functions
Definition 2.1 (Base Value). The information value of bet \(t\) in market \(m\) is the reduction in Kullback-Leibler divergence from the final state: \[v_{m,t} = D_{\text{KL}}(P_m^{\text{final}} \| P_{m,t-1}) - D_{\text{KL}}(P_m^{\text{final}} \| P_{m,t})\] where \(D_{\text{KL}}(P \| Q) = \sum_{i \in \mathcal{L}_m} P(i) \log\frac{P(i)}{Q(i)}\).
Definition 2.2 (Temporal Decay). The timeliness of prediction \(t\) is encoded via decay function \(\delta: [0, t_1] \to \mathbb{R}_+\): \[\delta(t) = \exp(-\lambda(t_1 - t)), \quad \lambda > 0\] or alternatively \(\delta(t) = 1 - t/t_1\) for linear decay.
Definition 2.3 (Normalized Value). Accounting for market frictions, the adjusted value is: \[v_{m,t}^{\text{adj}} = \delta(t) \cdot \frac{v_{m,t}}{c_{m,t}}\] where \(c_{m,t}\) represents the mechanism-specific cost of information at time \(t\).
3. Mechanism-Specific Cost Functions
Bayesian Markets. State represented by pseudo-counts \(\alpha_t \in \mathbb{R}_+^{|\mathcal{L}_m|}\) with update \(\alpha_t = \alpha_{t-1} + w \cdot q_t\). The cost of information is the prior strength: \[c_{m,t}^{\text{Bayes}} = \|\alpha_{t-1}\|_1 = \sum_{j \in \mathcal{L}_m} \alpha_{t-1}(j)\]
LMSR Markets. State represented by log-quantities \(q_t \in \mathbb{R}^{|\mathcal{L}_m|}\) with cost function \(C(q) = b \log \sum_i \exp(q_i/b)\). The cost is: \[c_{m,t}^{\text{LMSR}} = \nabla C(q_{t-1}) \cdot b_t = \sum_{i \in \mathcal{L}_m} P_{m,t-1}(i) \cdot b_t(i)\] where \(b_t\) is the share vector purchased.
4. System Value Aggregation
Definition 4.1 (Agent System Value). The total value of agent \(a \in \mathcal{K}\) is: \[V_{\text{system}}(a) = \sum_{m \in \mathcal{M}} \sum_{\substack{t=1 \\ \phi_m(t) = a}}^{T_m} \delta(t) \cdot \frac{D_{\text{KL}}(P_m^{\text{final}} \| P_{m,t-1}) - D_{\text{KL}}(P_m^{\text{final}} \| P_{m,t})}{c_{m,t}}\]
Proposition 4.2 (Additivity). System value is additive across agents: \[V_{\text{system}}(\mathcal{K}) = \sum_{a \in \mathcal{K}} V_{\text{system}}(a)\]
Proposition 4.3 (Counterfactual Equivalence). \(V_{\text{system}}(a)\) equals the expected KL divergence between the final outcome and the counterfactual market state without agent \(a\)’s participation.
5. Incentive Alignment
Let \(R(a)\) denote the reward function under mechanism \(\mathcal{M}\). The alignment gap is defined as: \[\Gamma(a) = V_{\text{system}}(a) - \mathbb{E}[R(a)]\]
Bayesian Markets. With scoring rule \(S(P, \omega)\) where \(\omega\) is the realized outcome: \[R_{\text{Bayes}}(a) = \sum_{m,t: \phi_m(t)=a} S(P_{m,t}, \omega_m) - S(P_{m,t-1}, \omega_m)\] The alignment gap depends on prior strength \(\|\alpha_0\|\) and decay parameter \(\lambda\).
LMSR Markets. With profit function \(\pi\): \[R_{\text{LMSR}}(a) = \sum_{m,t: \phi_m(t)=a} \pi_{m,t}\] The alignment gap depends on liquidity parameter \(b\) and timing \(t\).
6. Optimization Framework
The system designer solves: \[\max_{\lambda, b, \alpha_0} \sum_{a \in \mathcal{K}} V_{\text{system}}(a; \lambda, b, \alpha_0)\] subject to participation constraints \(\mathbb{E}[R(a)] \geq u_a^{\text{outside}}\) for all \(a \in \mathcal{K}\).
The Lagrangian yields optimal decay rate: \[\lambda^* = \frac{\sum_{a} \frac{\partial V_{\text{system}}(a)}{\partial \lambda}}{\sum_{a} \frac{\partial \mathbb{E}[R(a)]}{\partial \lambda}}\]
7. Conclusion
The framework provides a mechanism-agnostic metric \(V_{\text{system}}(a)\) for agent evaluation, incorporating temporal discounting \(\delta(t)\) and mechanism-specific costs \(c_{m,t}\). Future work concerns dynamic optimization of \((\lambda, b)\) to minimize the alignment gap \(\Gamma(a)\) across heterogeneous agent populations.