Introduction Model Results Discussion

Social Learning and the shadow of the Past Yuval Heller (Bar Ilan) and Erik Mohlin (Lund)

Stony Brook Conference on Game Theory, July 2017 In honor of Pradeep Dubey & Yair Tauman

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Motivation 1: Social Learning

Agents often make decisions without knowing the costs and benefits of the possible choices. A new agent may base his decision on observing the actions taken by a few incumbents. E.g., Arthur (1989, 1994); Ellison & Fudenberg (1993, 1995), Banerjee & Fudenberg (2004), Acemoglu, Dahleh, Lobel & Ozdaglar (2011), Sorensen & Smith (2014).

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Motivation 2: Games with Random Matching

Agents are randomly matched to play a game. An agent may base his choice of action on a few observations of how his current opponent behaved in the past. Application: community enforcement in the prisoner’s dilemma. E.g, Rosenthal (1979), Okuno-Fujiwara & Postlewaite (1995), Nowak & Sigmund (1998), Takahashi (2010), Heller & Mohlin (2017).

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Research Question

Question When does the initial behavior of the population have a lasting influence on the population’s behavior in the long run?

A broad class of models (including both social learning and games with random matching).

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Brief Summary of Main Results

If the expected number of observed actions is: < 1: The population converges to the same behavior independently of the initial state. > 1: There is a learning rule that admits multiple steady states, and the initial state determines which steady state will prevail.

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Example: Competing Technologies (1) 2 competing technologies, a & b with increasing returns. The initial share of agents following technology a is uniformly distributed in [30%, 70%]. Agents have small symmetric idiosyncratic preferences. In each period some agents are replaced with new agents. 99% of the new agents observe the technology of a single incumbent. Two cases for the remaining 1%: Case (I) observe nothing, and Case (II) observe three actions. Heller & Mohlin

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Example: Competing Technologies (2)

Nash equilibrium (unique if agents are not too patient): An agent observing a single incumbent mimics the incumbent’s technology. An agent observing three incumbents, mimics the majority. An agent observing nothing chooses a technology based on his own idiosyncratic preferences.

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Introduction Model Results Discussion

Motivation Research Question Brief Summary of Results Motivating Example

Example: Competing Technologies (3)

One cans show that: Case (I): Global convergence to 50%-50% Mean sample size = 0.99 < 1.

Case (II): convergence to everyone playing the action initially played by a majority of the agents. Mean sample size = 1.02 > 1.

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

Model

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

Population State Infinite population (continuum of agents with mass one). Time is discrete: 1, 2, 3, 4, 5, .... Each agent chooses an action. Each agent is endowed with a type. Population state: vector describing the aggregate distribution of actions played by each type of agents. Let Γ denote the set of all population states. Heller & Mohlin

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

New/Revising Agents

At each period a fixed share of the agents die and are replaced with new agents (or, alternatively, reevaluate their actions). The remaining agents continue to play the same action as they played in the past (inertia).

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

Samples of Observed Actions Each new agent observes a sample with a random size. We study two kinds of sampling methods: 1

Sampling actions of different random incumbents (e.g., social learning `a la Ellison & Fudenberg, 1995; Banerjee & Fudenberg, 2004).

2

Sampling a random type, and then sampling random actions played by agents of this type (possibly, past actions of the current partner, e.g., models of community enforcement `a la Nowak & Sigmund, 1998; Heller & Mohlin, 2017). Heller & Mohlin

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

Environment & Mean Sample Size

An environment is a tuple describing all aspects except the agent’s behavior All the above components described so far: set of actions, set of types, fraction of revising agents, sampling method, distribution of types, and distribution of sample sizes.

Let µl be the mean sample size, i.e., the expected number of actions observed by a random new agent. Heller & Mohlin

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

Stationary Learning Process

Learning rule – a function describing the behavior of a new agent as a function of the agent’s type and the observed sample. A (stationary) learning process is a pair consisting of an environment and a learning rule.

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Introduction Model Results Discussion

Population State Revising Agents and Samples Environment and Learning Rules Population Dynamics

Population dynamics

An initial state & a learning process determine a new population state. Let fP : Γ → Γ denote the mapping between population states induced by a single step of the learning process P. We say that γ ∗ ∈ Γ is a steady state if fP (γ ∗ ) = γ ∗ .

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Introduction Model Results Discussion

Main Result Additional Results and extensions

Theorem Let E be an environment. The following conditions are equivalent: 1

µl > 1 or all agents always observe a single action.

2

There is a learning rule, that admits 2 different steady states.

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Introduction Model Results Discussion

Main Result Additional Results and extensions

Theorem Let E be an environment. The following conditions are equivalent: 1

µl > 1 or all agents always observe a single action.

2

There is a learning rule, that admits 2 different steady states.

Sketch of Proof ¬1 ⇒ ¬2. We show that ¬1 implies that fp is a weak contraction (i.e.,

ψl,γ − ψl,γ 0 < kγ − γ 0 k ). Intuition: distance between new 1 1 population states ≤ distance between distributions of samples ≤ mean sample size * distance between old population states. ⇒ The unique steady state is a global attractor. Heller & Mohlin

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Introduction Model Results Discussion

Main Result Additional Results and extensions

Sketch of Proof 1 ⇒ 2 If all new agents observe a single action: Learning rule: each agent plays the observed action. Each initial state is a steady state.

If µl > 1, let the learning rule be such that each agent plays action a if he has observed action a at least once, and plays action b otherwise. Two steady states: Everyone plays b. A share x > 0 of the agents plays a, the rest plays b. Heller & Mohlin

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Introduction Model Results Discussion

Main Result Additional Results and extensions

Sketch of Proof 1 ⇒ 2

The learning rules in the proof can be consistent with Bayesian inference and best-replying. E.g., two technologies with unknown quality and two states of the world: 1

Technology a is better; initially 10% of the agents follow a.

2

Technology b is better; initially all agents follow technology b.

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Introduction Model Results Discussion

Main Result Additional Results and extensions

Additional Results and Extensions

Extending the main result to a setup with: 1

non-stationary learning processes that depend on the calendar time; and

2

random common shocks.

Presenting a tighter upper bound for a learning rule to admit a unique steady state based on the rule’s responsiveness to changes in the observed samples. Heller & Mohlin

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Introduction Model Results Discussion

Repeated Interactions without a Global Calendar Time Large Finite Populations Conclusion

Repeated Interactions without a Global Calendar Time Agents are randomly matched within a community, and these interactions have been going on since time immemorial. Arguably, these situations should be modeled as steady states of environments without a calendar time (e.g., Rosenthal, 79; Okuno-Fujiwara & Postlewaite 95; Phelan & Skrzypacz 06; Heller & Mohlin 17).

Is the distribution of strategies used by the players sufficient to uniquely determine the steady state? Our main result shows that this is true whenever the expected number of observed actions is less than one. Heller & Mohlin

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Introduction Model Results Discussion

Repeated Interactions without a Global Calendar Time Large Finite Populations Conclusion

Interpreting Our Results for Large Finite Populations µl < 1: each learning process admits globally-stable state γ ∗ . Quick convergence to γ ∗ , and almost always remaining close to γ ∗ .

µl > 1: there are learning rules with multiple steady states. Convergence in the medium run to one of the locally stable states, and staying there for a significant amount of time. Future research may rely on stochastic evolutionary stability analysis (Foster & Young, 90; Kandori et al., 93; Young, 93) to yield a long run prediction independent of the initial state. Heller & Mohlin

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Conclusion The initial population state has lasting influence on the long-run behavior essentially iff the mean sample size > 1: ≤ 1: Global convergence to a unique steady state. > 1: there are learning rules with multiple steady states. Extensions in the paper: (https://sites.google.com/site/yuval26/Research) Non-stationary environments. Environments with common random shocks. Tighter bounds that depend on the rule’s responsiveness.