Cross-Validation Equilibrium

This paper introduces and analyzes the concept of Cross-Validation Equilibrium, a strategic framework where players delegate belief formation to machine learning agents that select predictive models based on endogenous training data to minimize out-of-sample error, thereby influencing equilibrium outcomes in games like jury voting and speculative betting.

The Gist

Imagine a world where people don't just make decisions based on their own gut feelings or logic, but instead hire a "digital assistant" (a Machine Learning agent) to figure out what's going on. This is the core idea of the paper "Cross-Validation Equilibrium" by Ran Spiegler and Stephan Waizmann.

Here is a simple breakdown of what they are studying, using everyday analogies.

The Setup: The "Digital Assistant" Dilemma

In a normal game (like a business competition or a jury trial), players try to guess what others will do. In this paper, the players don't guess directly. Instead, they ask their AI assistant to build a model that predicts the outcome.

Here is the twist: The AI's training data comes from the players' own behavior.

Think of it like this:

• You are a chef trying to predict how many customers will show up.
• Your AI assistant looks at a "training sample" (a list of past customer numbers) to build a prediction model.
• The Catch: That list of past customer numbers was generated by other chefs who were also using their own AI assistants to decide how many customers to expect.
• It's a loop: Your AI learns from the world you and others created, and then you act based on what your AI tells you.

The Core Concept: "Cross-Validation Equilibrium" (CVE)

The paper introduces a new way to find a "stable state" (an equilibrium) in these games. They call it Cross-Validation Equilibrium.

The Analogy: The Student and the Pop Quiz
Imagine a student (the AI) trying to learn a subject.

1. Training: The student studies a specific set of notes (the training sample).
2. Testing: To see if they really understand, the student takes a "pop quiz" on new questions (the validation sample).
3. The Goal: The student wants to pick the study method (the model) that gets the best score on the pop quiz, not just the one that memorizes the notes perfectly.

In the paper's world:

• Players' AI agents look at different ways to group information (called "partitions").
• They pick the grouping that makes the fewest mistakes when predicting new data (out-of-sample).
• Once the AI picks the best grouping, the human player acts on that prediction.
• Because the data is random (noisy), the AI might pick a different grouping every time, leading to random behavior. The "Equilibrium" is the average result of all these random choices.

Key Findings (The "So What?")

The authors run this scenario through several games to see what happens. Here are the main takeaways:

1. Noise Makes Things Messy (and sometimes worse)
If the data the AI sees is "noisy" (full of errors or random fluctuations), the AI might decide that a "coarse" (simple) model is better than a "fine" (detailed) one.

• Analogy: If you are trying to predict the weather but your thermometer is broken and jittery, you might stop trying to predict the temperature for every hour and just guess "it's usually warm."
• Result: In business competition (like two firms setting prices), this noise can make prices and quantities swing wildly, even more than if everyone was acting rationally.

2. The "Team Effort" Paradox
In a game where players help each other (like a team project), the authors found something surprising: Multiple stable outcomes are possible.

• Analogy: Imagine two people trying to push a car. If they both think, "We need to push hard," they push hard. If they both think, "It's too hard, let's just coast," they coast.
• Result: Unlike standard game theory which usually predicts one clear outcome, this AI-driven setup can get stuck in a "low effort" loop or a "high effort" loop, depending on how the AI interprets the noisy data.

3. The "Illusion of Control"
In one specific example, the AI leads players to do something that is objectively bad for them.

• Analogy: Imagine a gambler who thinks their lucky shirt causes them to win. The data is random, but the AI finds a pattern where "wearing the shirt" and "winning" happened together by chance. The AI tells the player, "Wear the shirt!" and the player does, even though the shirt does nothing.
• Result: Players might take costly actions because their AI falsely believes those actions cause good outcomes.

4. Being "Smarter" Doesn't Always Win
In a betting game, the paper shows that a player who is forced to use the perfect, detailed model can actually lose money to a player who is allowed to use a "coarse" (simpler) model.

• Analogy: In a noisy room, the person trying to hear every single word (the detailed model) gets confused by the static. The person who just listens for the general vibe (the coarse model) actually understands the conversation better and wins the bet.
• Result: Sometimes, ignoring details (simplifying) is a better strategy when the data is messy.

Summary

The paper argues that when we delegate our decision-making to AI that learns from our own collective behavior, we enter a new kind of game. The AI tries to avoid "overfitting" (memorizing noise) by choosing simpler models, but this simplification can lead to:

• Wilder swings in market prices.
• Multiple possible outcomes for the same game.
• Players making mistakes they wouldn't make if they were thinking for themselves.
• Sometimes, "dumber" models winning against "smarter" ones.

It's a study of how the tools we use to understand the world can actually change the world in unexpected ways.

Technical Summary

Technical Summary: Cross-Validation Equilibrium

1. Problem Statement

This paper addresses the strategic implications of integrating predictive machine learning (ML) into static Bayesian games with incomplete information. In many economic settings (e.g., credit scoring, judicial risk assessment), human decision-makers delegate the formation of predictive beliefs to ML algorithms. These algorithms construct models based on random training samples drawn from the equilibrium distribution of the game itself.

The core problem is to characterize the equilibrium outcomes when:

1. Players delegate belief formation to an "ML agent."
2. The ML agent selects a predictive model (mapping player types to payoff-relevant outcomes) based on a noisy, endogenous training sample.
3. The model selection criterion mimics cross-validation: the agent chooses the model that minimizes expected out-of-sample squared prediction error.
4. Players then best-reply to the beliefs generated by the selected model.

The authors seek to understand how the endogenous link between players' behavior and the ML-generated predictive models alters standard equilibrium concepts like Nash Equilibrium (NE).

2. Methodology and Model Framework

The Game Structure

The authors formulate a slightly non-standard Bayesian game:

• Players: $N = \{1, \dots, n\}$.
• States and Types: A set of states $\Theta$ and type sets $T_i$.
• Payoffs: Player $i$'s utility $u_i(a_i, t_i, x_i)$ is continuous and linear in a scalar payoff-relevant outcome $x_i$. The outcome $x_i$ is a function of the state and opponents' actions, $x_i = f_i(\theta, a_{-i})$.
• Training Sample: For a fixed strategy profile $\sigma_{-i}$, the true expected outcome for player $i$ given type $t_i$ is $\bar{x}^\sigma_i(t_i)$. The ML agent observes a noisy training sample:
  $$y_{t_i} = \bar{x}^\sigma_i(t_i) + \varepsilon_{t_i}$$
  where $\varepsilon_{t_i}$ are i.i.d. noise terms symmetric around zero.

The ML Agent's Task

The ML agent selects a partition $\Pi_i$ of the type space $T_i$ from a feasible set $\mathcal{P}_i$.

• Prediction: Within each partition cell $\pi(t_i)$, the agent predicts the outcome using the weighted average of the training sample in that cell.
• Selection Criterion: The agent selects the partition that minimizes the expected out-of-sample mean squared prediction error. This involves a validation sample with independent noise $\eta$. The agent evaluates the partition's performance across many potential validation samples.
• Bias-Variance Trade-off: The selection process naturally balances bias (coarse partitions) and variance (fine partitions). A coarse partition may be selected if the training sample noise is high, as it reduces the variance of the estimator.

Definition: Cross-Validation Equilibrium (CVE)

A strategy profile $\sigma$ is a Cross-Validation Equilibrium (CVE) if, for every player $i$ and every realization of the training sample $\varepsilon_i$:

1. The ML agent selects a partition $\Pi_i$ that minimizes the expected out-of-sample prediction error given $\varepsilon_i$ and the opponents' strategies.
2. The player chooses a pure strategy $g_i(t_i)$ that is a best-reply to the belief induced by the selected partition $\Pi_i$.
3. The player's mixed strategy $\sigma_i$ is the mixture over all such compatible (partition, strategy) pairs, weighted by the probability of the training sample realization.

3. Key Theoretical Results

Existence and Convergence

• Existence: The authors prove that a CVE exists in every finite game using a fixed-point argument on a correspondence mapping strategy profiles and partition distributions to themselves.
• Convergence to ABEE: As the training sample noise vanishes ($\text{Var}(\varepsilon) \to 0$), any limit point of a sequence of CVEs converges to an Analogy-Based Expectation Equilibrium (ABEE) (Jehiel, 2005) with respect to the finest feasible partition.
• Convergence to NE: If the finest feasible partition is the singleton partition (distinguishing every type), the CVE converges to the Nash Equilibrium as noise vanishes.

Properties of Model Selection

• Unbiasedness of Selected Models: A crucial technical result (Proposition 4) is that the expected training sample noise conditional on the selected partition is zero. This holds because the selection criterion (minimizing expected out-of-sample error with a quadratic loss) and the symmetry of the noise distribution ensure that the selection process does not systematically bias the noise.
• Avoidance of Over-fitting: If the sufficient statistic is measurable with respect to a coarse partition, the ML agent will almost never select a strictly finer partition, as the finer partition would only increase variance without reducing bias (Proposition 3).
• Partition Selection Rule: In binary settings (fine vs. coarse partitions), the fine partition is selected if and only if the variance of the realized training sample noise is less than the variance of the sufficient statistic (Proposition 5).

4. Applications and Economic Implications

The paper applies CVE to several specific games to illustrate qualitative differences from Nash Equilibrium:

A. Quadratic-Payoff Games (Cournot and Team Effort)

• Strategic Substitutability (Cournot): In a Cournot competition, the volatility of equilibrium quantities and prices in a symmetric CVE increases with the variance of the training sample noise. This occurs because noisier samples make coarse partitions more likely; under strategic substitutability, coarse beliefs (which under-react to state fluctuations) lead to amplified reactions by the player.
• Strategic Complementarity (Team Effort): In a team effort game, the CVE can exhibit multiple equilibria even when the underlying game has a unique Nash equilibrium. The multiplicity arises because the probability of selecting a fine partition is increasing in the volatility of beliefs, which in turn depends on the partition selection.
• Discontinuity: Even with arbitrarily small noise, the CVE in the team effort game can be discontinuously different from the Nash equilibrium, yielding lower expected payoffs.

B. Unanimity Voting (Jury Model)

In a two-player jury model, endogenous model selection can lead to higher probabilities of voting for a reform compared to the efficient Nash equilibrium. The noise in the training sample can induce the selection of a coarse partition, which aggregates information in a way that makes players more likely to vote "Yes" than they would under rational, fine-grained beliefs.

C. Speculative Betting

In a zero-sum betting game, a player restricted to the "correct" fine partition may suffer lower expected payoffs than an opponent who has access to both fine and coarse partitions. The opponent can strategically select a coarse partition when noise realizations are opposite, effectively averaging out the noise to form a correct belief and exploit the restricted player. This challenges the intuition that more information (or the correct model) is always superior.

D. Illusion of Control

The authors extend the framework to allow partitions of the joint space of states and actions. They demonstrate a scenario where players choose a strictly dominated action with positive probability. This "illusion of control" arises because the ML agent occasionally selects a model that falsely attributes a successful outcome to the player's own action, leading the player to believe the action is causal.

5. Significance and Contribution

The paper claims to contribute to the literature on:

1. Strategic ML: Unlike reinforcement learning literature (e.g., Calvano et al., 2020) which focuses on long-run learning dynamics, this paper focuses on static games where ML agents construct predictive models for human players.
2. Non-Rational Belief Formation: It extends the literature on subjective equilibrium concepts (e.g., Jehiel, 2005; Eyster and Rabin, 2005) by introducing endogenous, random model selection. Unlike previous models where partitions are fixed or exogenously given, CVE features partitions selected endogenously based on a cross-validation criterion.
3. Endogenous Data: It highlights that when training data is endogenous to the equilibrium, the "bias-variance trade-off" inherent in ML model selection becomes a strategic variable, potentially leading to multiple equilibria, increased volatility, and outcomes that deviate significantly from rational benchmarks even with small amounts of noise.

The authors emphasize that their results rely on the specific criterion of minimizing expected out-of-sample squared error (cross-validation) and the symmetry of noise. They note that the linearity of payoffs is a simplifying assumption that allows the sufficient statistic to be a scalar, facilitating the link to ABEE.