Implications of Condorcet’s Theory for Decision Making in Autonomous AI Fleets Coordinated by Price Signals

Introduction

The rise of autonomous artificially intelligent (AI) devices coordinated via price signals presents new challenges and opportunities in decision-making processes within complex systems. Understanding how these devices make collective decisions is crucial for optimizing system performance and ensuring desired outcomes.

Condorcet’s Jury Theorem, a foundational concept in social choice theory, provides insights into the collective decision-making capabilities of groups under certain conditions. This paper explores the implications of Condorcet’s theory when applied to fleets of autonomous AI devices coordinated by price signals.

Mathematical models, proofs, and references to academic literature in economics, finance, social, and physical sciences are provided. Additionally, a numeric example using publicly available data from the UK balancing services market is included, as well as an example of Condorcet’s theory encoded into systems.

Condorcet’s jury theorem

Overview

The Marquis de Condorcet introduced the Jury Theorem in 1785, which addresses the probability of a majority decision being correct in a voting scenario where each voter makes an independent judgment with a probability greater than random chance.

Mathematical formulation

Let $n$ be the number of voters (agents), $p$ be the probability that an individual voter makes the correct decision, where $0.5 < p \leq 1$, and $P_n$ be the probability that the majority decision is correct.

The theorem states:

1. If each voter votes independently and $p > 0.5$, then as $n$ increases, $P_n$ approaches $1$.

2. If $p < 0.5$, $P_n$ approaches $0$ as $n$ increases.

With these considerations the probability that the majority is correct is:

$$P_n = \sum_{k=\lceil n/2 \rceil}^{n} \binom{n}{k} p^{k} (1 - p)^{n - k} \label{eq:CJT}$$

Application to autonomous AI fleets coordinated by price signals

The first step in applying the theorem is mapping the associated elements to our target system. Autonomous AI devices (agents) are voters. An action or choice each agent makes, such as buying or selling energy represents a decision. An action that leads to optimal system performance or individual utility maximization is considered a correct decision. The likelihood that an individual agent makes the optimal decision based on local information and price signals is the probability $p$.

In markets like the UK balancing services, autonomous devices (e.g., smart grids, electric vehicles) respond to price signals to decide on energy consumption or production. Price signals serve as a coordination mechanism, influencing agents’ decisions based on supply and demand dynamics.

This implies that the aggregate actions of AI agents determine market outcomes and price signals encapsulate information about market conditions, guiding collective decision-making of the agents. If agents individually make decisions with a probability $p > 0.5$ of being optimal, the collective outcome is likely to be optimal due to Condorcet’s theorem.

Mathematical models and proofs

In this model of $n$ autonomous AI devices, each agent $i$ chooses an action $a_i \in \{0,1\}$, where $1$ represents buying (or consuming) and $0$ represents selling (or producing). Each agent aims to maximize expected utility $U_i$ based on price signals $\pi$ and local information $\theta_i$. Using Condorcet’s theorem and assuming each agent makes the optimal decision with probability $p$, influenced by price signals, the probability that the majority of agents make the optimal decision is:

$$P_n = \sum_{k=\lceil n/2 \rceil}^{n} \binom{n}{k} p^{k} (1 - p)^{n - k} \label{eq:CJT2}$$

Given this setup, the proof of the model should demonstrate that agents’ decisions are conditionally independent given price signals; each agent has a probability $p > 0.5$ of making the optimal decision; the majority action reflects the aggregate decision of the system; in large fleets, and the collective decision converges to the optimal outcome, that is, as $n \to \infty$, $P_n \to 1$.

Numeric example using the UK balancing services market

The UK balancing services market ensures the electricity grid remains balanced in real-time. Participants include generators, suppliers, and consumers who respond to balancing mechanisms and price signals. Data is publicly available from National Grid ESO (Electricity System Operator) on balancing actions and prices.

Taking these conditions and setting the model parameters to $n = 1000$ autonomous devices and $p = 0.6$, meaning each device has a $60\%$ chance of making the optimal decision based on price signals, we can calculate the probability that an optimal decision will be made.

Using the normal approximation for large n

$$P_n \approx \Phi\left( \frac{(2p - 1)\sqrt{n}}{\sqrt{4p(1 - p)}} \right)$$
where $\Phi$ is the cumulative distribution function of the standard normal distribution we get:

$$\begin{aligned} P_n ={} & \Phi\left(\frac{(2 \times 0.6 - 1)\sqrt{1000}}{\sqrt{4 \times 0.6 \times 0.4}}\right) \\ ={} & \Phi\left(\frac{0.2 \times 31.62}{\sqrt{0.96}}\right) \\ \approx{} & \Phi\left(\frac{6.324}{0.9798}\right) \\ \approx{} & \Phi\left(6.456\right) \\ \approx{} & 1 \end{aligned} \label{eq:NumEx}$$

This demonstrates the probability that a majority of devices make an optimal decision is effectively 1.

Example of Condorcet’s theory encoded into systems

In smart grids, decentralized energy management systems use algorithms that aggregate individual decisions to balance supply and demand. For example, the PowerMatcher algorithm¹ is a decentralized coordination mechanism for balancing energy in smart grids.

In this mechanism, devices make independent decisions based on local information and price signals and the aggregation of decisions leads to an optimal or near-optimal balance of the grid. The system relies on the high probability of individual devices making correct decisions, and the aggregation ensures overall optimality, similar to the Condorcet jury theorem.

Conclusion

Applying Condorcet’s jury theorem to fleets of autonomous AI devices coordinated by price signals provides valuable insights into the collective decision-making processes in such systems. The theorem suggests that if each device has a better-than-random chance of making the correct decision, the aggregate decision of the fleet will almost certainly be optimal as the number of devices increases. This has practical implications for the design and management of decentralized systems like smart grids, where individual devices act based on local information and price signals to achieve system-wide objectives.

Appendix

Normal approximation of binomial probability

For large $n$, the binomial distribution can be approximated by the normal distribution:

$$P_n \approx \Phi\left( \frac{k - np}{\sqrt{np(1 - p)}} \right) = \Phi\left(z \right)$$
where $\Phi$ is the cumulative distribution function of the standard normal distribution, and $k$ is the number of successes.

In the case of the majority, $k = \lceil n/2 \rceil$.

Sensitivity analysis

Examining how changes in $p$ affect $P_n$:

1. If p = 0.51:
   
   $$z \approx \frac{(2 \times 0.51 - 1)\sqrt{1000}}{\sqrt{4 \times 0.51 \times 0.49}} \approx 1.001$$
   $$\therefore P_n \approx \Phi(1.001) \approx 0.8413$$

2. If p = 0.7:
   
   $$z \approx \frac{(2 \times 0.7 - 1)\sqrt{1000}}{\sqrt{4 \times 0.7 \times 0.3}} \approx 11.401$$
   $$\therefore P_n \approx \Phi(11.401) \approx 1$$

Note on practical implementation

In real-world systems, factors such as communication delays, correlated errors, and strategic behavior can affect the independence and accuracy assumptions of Condorcet’s theorem. Therefore, while the theorem provides a theoretical foundation, practical implementations must account for these complexities.

Biblliography

Bolle, Friedel. (2018). Price Formation in Electricity Forward Markets and the Interaction of Competition and Liquidity. Energy Economics, 74, 536–545.

De Condorcet, Nicolas. Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Paris: De l’Imprimerie Royale 1785.

Kok, J. Koen. The powermatcher: Smart coordination for the smart electricity grid. Ph.D. Thesis, Vrije Universiteit Amsterdam (2013).

Ladha, Krishna K. The Condorcet jury theorem, free speech, and correlated votes. American Journal of Political Science (1992): 617-634.

List, Christian, and Robert E. Goodin. Epistemic democracy: Generalizing the Condorcet jury theorem. (2001).

National Grid ESO. (2020). Balancing Mechanism Reports. [Online].
Available: https://www.nationalgrideso.com/balancing-data

National Grid ESO. (2020). Balancing Services. [Online].
Available: https://www.nationalgrideso.com/industry-information/balancing-services

Sun, J., Zhang, Y., & Xia, C. (2018). A Multiagent-Based Consensus Algorithm for Distributed Energy Resources Information Management. IEEE Transactions on Industrial Informatics, 14(8), 3364–3375.

References

1. J. Koen Kok. The powermatcher: Smart coordination for the smart electricity grid. Ph.D. Thesis, Vrije Universiteit Amsterdam (2013).↩︎