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Evolutionary Approach to �Multi-dimensional Learning �with Application to Firms

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Srinivas Arigapudi (IIT Kanpur), Omer Edhan (Manchester),

Yuval Heller & Ziv Hellman (Bar-Ilan University)

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�BIU, October 2025

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Outline

  • Motivation and summary
  • Basic setup and definitions
  • Benchmark: Replicator dynamics
  • Combinator dynamics (learning one dimension at a time)
  • Family of the recombinator dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Motivation and Brief Summary

  • The replicator dynamic (Taylor & Jonker, 1978) is a central model �in evolutionary game theory. Social learning interpretation: mimicking the behavior of a successful incumbent (mentors)
  • People think of complex strategies as multi-dimensional:

Arad & Rubinstein (2018): “in games with a large and complex strategy space, players tend to think in terms of strategy characteristics rather than the strategies themselves; in their strategic deliberation, players consider one characteristic at a time.”

  • We present a tractable family of recombinator dynamics, in which some agents learn one strategic dimension at a time
  • Main result: Characterize stationary states and stable states (predictions of long-run behavior)
  • Main application: Behavior in firms

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Outline

  • Motivation and brief summary
  • Basic setup and definitions
  • Benchmark: Replicator dynamics
  • Combinator dynamics
  • Family of the recombinator dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Basic Setup

 

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Continuous-Time Dynamics

 

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Strategic Dimensions

 

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Example 1: Coordinated Improvement

  • Firm faces binary choices in two domains: marketing and product design
  • In each domain: either conventional (c) or innovative (i)
  • 2 dimensions * 2 actions
  • For simplicity, firm’s payoff is independent of other firms:

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  • Difficult for CEO to consider all dimensions simultaneously (see, e.g., Gibbons & Henderson, 2013)
  • Each division chooses its trait separately
  • The firm combines these chosen traits into its new action

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Definitions: Stationarity and Stability

 

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Outline

  • Motivation and brief summary
  • Basic Setup and definitions
  • Benchmark: Replicator dynamics
  • Combinator dynamics
  • Family of the Recombinators dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Benchmark: Replicator Dynamics

 

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Stationary States in the Replicator Dynamics

Known results:

  • The replicator dynamics is payoff-monotone: �actions with higher payoffs become more frequent
  • State is stationary iff all incumbent actions have the same payoffs
  • Any Nash equilibrium is a stationary state
  • Any full-support stationary state is a Nash equilibrium

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Stable States in the Replicator Dynamics

 

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Outline

  • Motivation and brief summary
  • Basic setup and definitions
  • Benchmark: replicator dynamics
  • Combinator dynamics (learning one dimension at a time)
  • Family of the recombinator dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Combinator Dynamics

 

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Combinator Dynamics: Payoff Monotonicity

 

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Combinator Dynamics: Stationary States

 

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Combinator Dynamics: Stable States

 

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Stable States – Example 1

  • Replicator dynamics (CEO make coordinated decisions):
    • unique globally-stable action: ii
  • Combinator dynamics (independent decisions in each domain):
    • 2 locally-stable pure states: ii and cc�(+ Unstable heterogenous state)

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Example 2: Synergistic Congestion Game

  • Firm faces binary choices in marketing and product design
  • Each division can either focus on young consumers (y) or old (o)
  • 3 components to the firm’s payoff:
    • Marketing-congestion / design-congestion: Higher payoff if fewer firms choose the same trait as the firm in that domain
    • Across-domain synergy: Additional payoff if the two divisions focus on the same type

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Example 2: Synergistic Congestion Game

 

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Outline

  • Motivation and brief summary
  • Basic setup and definitions
  • Benchmark: replicator dynamics
  • Combinator dynamics
  • Family of the recombinator dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Outline

  • Motivation and brief summary
  • Basic setup and definitions
  • Benchmark: Replicator dynamics
  • Combinator dynamics
  • Family of the Recombinators dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Payoff Monotonicity

 

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Characterization of Stationary States

 

Mentors & Recombinators

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Characterization of Stable States

 

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Formal definition

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Example 2 (Revisited): Synergistic Congestion Game

 

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Outline

  • Motivation and brief summary
  • Basic Setup and definitions
  • Benchmark: Replicator dynamics
  • Combinator dynamics
  • Family of the Recombinators dynamics
  • Results
    • Payoff monotonicity
    • Stationary states
    • Asymptotically stable states
  • Discussion & conclusion

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Insights

  • Persistent performance gaps between
    • Dominated actions might survive.
  • An action with a higher payoff:
    • Positive correlation between its composing traits.
    • Its traits have lower payoffs when combined with other traits.
  • All traits in a stable state have the same (maximal) payoff.
  • Additional insight in the paper:
    • Lower recombination rates benefit specialized traits �(high payoffs when combined with specific traits)
    • Higher recombination rates benefit versatile traits who work relatively well with most traits in the other dimensions.

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Related Literature (1)

  • Multi-dimensional reasoning in games:
    • Arad & Rubinstein (2012, 2019), Arad & Penzynski (2018)
    • A static solution concept (multi-dimensional equilibrium) similar to our notion for pure equilibria in the combinator dynamics
  • Decision-making in firms:
    • Persistent performance differences: Bartelsman & Doms (2000), Syverson (2004, 2011); Hsieh & Klenow (2009)
    • Variation in management practices (Gibbons & Henderson, 2012)
    • Rugged performance landscape (Levinthal, 1997)
    • Difficulties coordinated decisions accorss divisions in a firm (Milgrom & Roberts, 1995; Rivkin, 2000; Brynjolsson & Milgrom, 2013)
    • Recombinant innovation (Henderson & Clark, 1990; �Weitzman, 1998; Griffith, Lee & Straathos, 2017)

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Related Literature (2)

  • Replicator dynamics (Taylor & Jonkers, 1978; Borgers & Sarin, 1997; Hopkins, 2002; Cressman & Tao, 2014; Metikopoulos & Sandholm, 2018)
  • Sexual genetic inheritance:
    • Biological literature: Karlin (1975), Eshel & Feldman (1984), �Matessi & DI Pasquale (1996).
    • Economic literature, focusing on pure stable states & biases compensate each other: Waldman (1994), Frenkel, Heller & Teper (2018).
    • Relations with learning algorithms: Chastain, Livnat, Papadimitriou & Vazirani (2014), Barton, Novak & Paixao (2014), Meir & Parks (2015), Edhan, Hellman & Sherill-Rofe (2017), Palaiopanos, G., I. Panageas, and G. Piliouras (2017), and Edhan, Hellman & Nehama (2021).
  • Key contribution: Existing literature mainly focuses on non-strategic situations (player's payoff are independent of the actions chosen by others in the population).

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Conclusion

 

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Backup Slides

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Formal Definition of Post-Invasion Payoff

 

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