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Capturing the Complexity of Human Strategic Decision-Making with Machine Learning

Jian-Qiao Zhu, Joshua C. Peterson, Benjamin Enke, Thomas L. Griffiths

arXiv:2408.07865v1econ.GNcs.GTcs.LGq-fin.EC

TL;DR

Explaining human strategic choices is difficult because people often violate the mutual-belief and rational-response assumptions of Nash equilibrium. The paper compares behavioral models with neural predictions and builds game-dependent neural augmentations of a level-k quantal-response model. Context-dependent models outperform context-invariant ones, and the authors link variation in responding and reasoning to game complexity.

  • Problem

    Human players often violate Nash equilibrium’s assumptions of mutually consistent beliefs and mutual rationality, limiting its effectiveness in explaining strategic behavior.

    Source: p. 2

  • Method

    The authors compare behavioral models with neural predictions and integrate neural networks into a level-k quantal-response model to make its structural parameters game-dependent.

    Source: pp. 4, 5, 7

  • Results

    Context-dependent models outperform context-invariant models; replacing all three structural parameters with neural networks reaches 97% completeness, versus 96% for a model with fixed k = 2 and context-dependent ηself and ηother.

    Source: pp. 4, 7

  • Takeaways & Limitations

    The authors interpret game-specific variation as reflecting how game complexity conditions players’ ability to optimize responses and reason about others, and validate a complexity index against independent markers.

    Source: p. 7

Abstract

from arXiv · show

Understanding how people behave in strategic settings--where they make decisions based on their expectations about the behavior of others--is a long-standing problem in the behavioral sciences. We conduct the largest study to date of strategic decision-making in the context of initial play in two-player matrix games, analyzing over 90,000 human decisions across more than 2,400 procedurally generated games that span a much wider space than previous datasets. We show that a deep neural network trained on these data predicts people's choices better than leading theories of strategic behavior, indicating that there is systematic variation that is not explained by those theories. We then modify the network to produce a new, interpretable behavioral model, revealing what the original network learned about people: their ability to optimally respond and their capacity to reason about others are dependent on the complexity of individual games. This context-dependence is critical in explaining deviations from the rational Nash equilibrium, response times, and uncertainty in strategic decisions. More broadly, our results demonstrate how machine learning can be applied beyond prediction to further help generate novel explanations of complex human behavior.

Supplementary Information

The supplementary materials detail how the study generated its games, recruited participants, and ran the experiments. They also describe the follow-up procedure and preregistration.

Game generation algorithm

  • The dataset contains 2,416 game instances generated from 1,208 payoff matrices, sampled across 142 game types with fewer dominance-game instances.Dominance games made up 87.5% of game types; the procedure generated three instances per double-dominance game, eight per single-dominance game, and 22 per non-dominance game.Source: Game generation algorithm · p. 13

Participants

  • Of 4,900 recruited U.S. participants, 4,673 completed the 10-minute experiment; participants played 20 one-shot games as the row player.Participants passed a game-understanding quiz, received no practice or coaching, and were anonymously rematched with a new opponent for each game without performance feedback.Source: Participants · p. 13, Procedure · p. 14

Follow-up Experiment

Games

  • The follow-up experiment used the identical game-generation algorithm from the main experiment to create 500 new normal-form games, maintaining similar proportions of dominance games as observed in the main experiment.Source: Games · p. 14

Participants

The follow-up recruited a new participant sample and added confidence judgments after strategic choices. The study also aggregated choices, response times, and confidence into game-level measures.

Participants

  • Of 1,013 recruited participants, 1,008 completed the follow-up experiment, which added a post-choice confidence rating to the original procedure.Participants used a 0%–100% slider to report certainty that their chosen option was their best decision.Source: Participants · p. 14, Procedure · p. 15

Data Preprocessing

  • The study collected strategic choices, response times, and confidence, then derived game-level measures from trial-level data.Choice frequencies across participants were the model-fitting targets; response times were log-transformed and standardized within participants before taking the game-level median, while confidence was standardized within participants and averaged by game.Source: Data Preprocessing · p. 15

Context-invariant Models

These context-invariant models use parameters that stay fixed across games, while specifying different accounts of strategic reasoning and noisy choice.

Context-invariant Models

  • Pure- and mixed-strategy Nash equilibria require that no player can improve by unilaterally changing, respectively, a single action or a probability distribution over actions.In both cases, each player’s strategy is a best response to the other players’ strategies.Source: Context-invariant Models · p. 16
  • The level-k model represents strategic sophistication as a hierarchy in which each level best responds to beliefs about players one level below.Level-0 players act without strategic consideration; level-1 players best respond to expected level-0 actions.Source: Context-invariant Models · p. 16
  • Quantal response models represent choice as probabilistic: strategies with higher expected payoffs are more likely, but suboptimal choices remain possible.The account contrasts with perfect best responding in Nash equilibrium.Source: Context-invariant Models · p. 16
  • The level-k quantal-response model combines level-k beliefs with noisy responses, with each level expecting lower-level players to use the same quantal-response logic.The reasoning chain continues down to random level-0 actions.Source: Context-invariant Models · pp. 16–17
  • The belief-noise extension lets players use separate quantal-response functions for their own choices and for the choices they believe opponents make.The passage notes evidence that people may underestimate opponents’ rationality.Source: Context-invariant Models · p. 17

Context-dependent Models

Context-dependent models use an MLP to turn each game matrix into parameters that vary across games. The paper also describes a direct-prediction MLP and neural-network augmentations of behavioral models.

Context-dependent Models

  • A context-dependent model uses an MLP to process each game matrix and generate corresponding model parameters.This allows parameter values to change with the game matrix.Source: Context-dependent Models · p. 17
  • The study’s MLP architecture has three hidden layers of 300 neurons, using sigmoid activations except for probability outputs, which use softmax.The highest-performing model on games2p2k directly predicts choice probabilities from the game matrix.Source: Context-dependent Models · p. 17
  • Neural-network augmentations make strategic sophistication, players’ own response noise, and believed opponent noise game-dependent inputs to a quantal-response function.Because k is discrete, the model represents sophistication as a distribution over levels k = 0, 1, 2, 3.Source: Context-dependent Models · p. 17

Optimization and Cross-validation

Models were optimized against game-level choice proportions, with neural and context-invariant models using different data splits and optimization procedures. Both model families were evaluated across ten repeated cross-validation rounds.

Optimization and Cross-validation

Developing a Game Complexity Index

The complexity index was developed by using the context-dependent model’s self-noisiness parameter as a target and selecting informative game features with LASSO regression.

Developing a Game Complexity Index

Game Topology

The Robinson and Goforth topology organizes 2 × 2 matrix games through the ordinal payoff rankings of each player.

Game Topology

  • Combining each player’s 12 unique payoff order graphs yields a topology of 144 ordinal order-graph pairs for 2 × 2 games.Source: Game Topology · pp. 18–19
  • The row-player examples include Chicken, Leader, Hero, Compromise, Deadlock, Prisoner’s Dilemma, Stag Hunt, Assurance, Safe Coordination, Peace, Harmony, and Concord.Source: Game Topology · pp. 18–19
  • The example matrix distinguishes the row player’s payoffs in blue from the column player’s payoffs in red.Source: Game Topology · p. 19

Game Features

The section describes game-matrix features intended to characterize complexity, spanning strategic solvability, payoff structure, and equilibrium properties.

Game Features

  • Dominant solvability marks whether a player’s game is solvable by iterated deletion of dominated actions, separately for the row and column players.Source: Game Features · p. 19
  • Excess dissimilarity extends a risky-choice measure of value-aggregation difficulty to matrix games, assuming the opponent’s strategies are random.Source: Game Features · p. 20
  • Levels of iterative rationality measure the minimum best-response iteration at which dynamics converge, capped at three for 2 × 2 games.Source: Game Features · p. 20
  • The features count pure- and mixed-strategy Nash equilibria and distinguish equilibrium payoff dominance from payoff dominance by a non-equilibrium outcome.Source: Game Features · p. 20
  • Nash equilibrium Pareto dominance records whether a pure-strategy equilibrium yields better payouts than all other such equilibria, treating a sole equilibrium as dominant.Source: Game Features · p. 20
  • Pure motives describe games where players’ choice preferences have perfect rank correlation, whether positive in coordination or negative in zero-sum games.Source: Game Features · p. 20
  • Payoff features include each player’s maximum payout and payout variance, plus a measure of deviations from zero-sum structure across outcomes.Source: Game Features · pp. 20–21
  • Payout inequality measures disparity in players’ maximum possible gains, while payout asymmetry measures deviations from a symmetric game.Source: Game Features · p. 21

Expected Relationships Between Game Features and Complexity

The proposed game features are grouped by how they are expected to relate to game complexity. They cover strategic demands, coordination, perspective-taking, and cognitive processing.

Expected Relationships Between Game Features and Complexity

  • Games are less complex when some strategies are never chosen under any best response, while Dominant Solvability requires consistent beliefs and best responses.Dominant Solvability captures the need for consistency in beliefs about opponents and the ability to best respond to those beliefs.Source: Expected Relationships Between Game… · pp. 21–22
  • Features that facilitate coordination are expected to reduce complexity when equilibrium and efficient outcomes align, and increase it when they diverge.The listed features include Nash Equilibrium Payoff and Pareto Dominance, Maximum Payouts, and Non-Nash Equilibrium Payoff Dominance.Source: Expected Relationships Between Game… · p. 22
  • Games are expected to be less complex when players can easily adopt an opponent’s perspective, including when preferences are perfectly rank-correlated positively or negatively.Pure Motives, Deviation from Zero-Sum Games, Inequality in Payouts, and Asymmetry in Payouts capture this property.Source: Expected Relationships Between Game… · p. 22
  • Multiple equilibria, variable expected-value computations, and iterative reasoning are expected to raise complexity when they demand greater cognitive resources.The cited features include the number of pure and mixed strategy Nash equilibria, Excess Dissimilarity, Payoff Variance, and Levels of Iterative Rationality.Source: Expected Relationships Between Game… · p. 22

Selection of Game Features

Decision-tree regression highlights three game features associated with strategic choices, while LASSO regression is used to construct a game-complexity index for explaining choice noisiness.

Selection of Game Features

  • A depth-3 decision tree ranked payoff-dominance by a Nash Equilibrium as the most important feature, followed by payoff dissimilarity and iterative rationality.The third layer repeated features from the second, leading the authors to identify these three as critical for strategic choices.Source: Selection of Game Features · p. 22
  • The game-complexity index was designed to explain noisiness in strategic choices and encapsulate deviations from rationality observed in the games.The LASSO regression used normalized game features to predict the negated ηself parameter from the best-fitting level-2 quantal response model, with an L1 multiplier of 0.2.Source: Selection of Game Features · pp. 22–23
  • The LASSO regression results are summarized in Table S4.Source: Selection of Game Features · p. 23

Robustness Check

The authors tested whether models trained on the main experiment generalized to follow-up data, and report that the main results were replicated.

Robustness Check

  • The main results were replicated when context-invariant and context-dependent models trained on main-experiment data were tested on follow-up data.Source: Robustness Check · p. 23
  • Context-invariant models were trained on the entire main-experiment dataset, while context-dependent models used a 90/10 training-validation split and optional stopping.Source: Robustness Check · p. 23
  • The follow-up experiment served as the test set for both model types.Source: Robustness Check · p. 23

Experimental Instructions and Comprehension Checks

The supplementary materials document participant instructions, trial procedures, and supporting model and feature analyses. They also include screenshots of experiment materials and tables describing model comparisons and regression results.

Experimental Instructions and Comprehension Checks

  • Supplementary screenshots show the main experiment’s comprehension check and game-matrix trial, alongside follow-up instructions and a trial with a confidence rating.Source: Figure S2, Figure S3, Figure S4
  • The model-comparison table defines completeness using averaged MSE and R2, except for the Nash equilibrium model, which uses R2 alone.Neural-network components are colorized, and parenthetical values are standard errors from 10-fold cross-validation.Source: Experimental Instructions and… · p. 25
  • The feature table reports expected correlations with game complexity, with negative and positive correlations indicating decreases and increases, respectively.Source: Experimental Instructions and… · p. 26
  • The supplement includes LASSO regression results and a table comparing models trained on the main experiment and tested on the follow-up experiment.The follow-up comparison table reports standard errors from 10-fold cross-validation.Source: Experimental Instructions and… · pp. 27–28
  • Participants played 20 matrix games, choosing a row while columns represented the other player’s options and payoffs were displayed in contrasting colors.The main-experiment instructions described payoff dependence on both players’ choices and illustrated the game matrix.Source: Experimental Instructions and… · pp. 29, 31
  • Participants were asked how certain they were that their chosen decision was actually their best decision given their preferences, expressing the likelihood in percent.The response scale showed 0%, 25%, 50%, 75%, and 100%.Source: Experimental Instructions and… · pp. 31–32
  • Participants were randomly rematched for each game, received no performance feedback during the session, and were paid using a randomly selected game plus base compensation.Source: Experimental Instructions and… · p. 31
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