Introduction Model Main Results Extensions and Discussion
Social Learning and the Shadow of the Past Yuval Heller (Bar Ilan) and Erik Mohlin (Lund)
Technion, Game Theory seminar, November. 2017
Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Motivation 1: Social Learning
Agents often make decisions without fully knowing costs & benefits. A new agent may learn from the experience of others, by basing his decision, on observations of actions taken by a few old agents. E.g., Arthur (1989, 1994); Young (1993); Kandori, Rob & Mailath (1993); Ellison & Fudenberg (1993, 1995), Banerjee & Fudenberg (2004), Acemoglu, Dahleh, Lobel & Ozdaglar (2011), Sorensen & Smith (2014).
Question When does the initial behavior of the population have a lasting effect? Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and 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. E.g, Community enforcement in the Prisoner’s Dilemma (Rosenthal, 1979; Okuno-Fujiwara & Postlewaite, 1995; Nowak & Sigmund, 1998; Takahashi, 2010; Heller & Mohlin, 2017).
Question When does the initial behavior of the population have a lasting effect? Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Modeling Approach & Research Question We divide the description of the interaction into two parts: 1
Learning rule: How new agents choose their actions (and keep playing the same actions throughout their lifetimes).
2
Environment: All other aspects: what agents observe, number of feasible actions, heterogeneity in the population, etc.
Question In which environments can the initial behavior of the population have a lasting effect? Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Brief Summary of Main Results
Average number of actions
≤1
∈ (1, 2]
>2
observed by a new agent Is there a rule with multiple steady states? Is there a rule with multiple locally stable states?
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Brief Summary of Main Results
Average number of actions
≤1
∈ (1, 2]
>2
observed by a new agent Is there a rule with
No
multiple steady states?
(global
Is there a rule with multiple
convergence)
Yes
Sometimes
Yes
locally stable states?
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Example: Competing Technologies (Environment) 2 competing technologies, a & b with increasing returns. Uncertainty about the initial share of agents following technology a (e.g., it is uniformly distributed in [0, 1]). 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 1 incumbent. Two cases for the remaining 1%: Case (I) observe nothing, and Case (II) observe three actions. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Example: Competing Technologies (Learning Rule)
Learning rule: An agent observing a single incumbent mimics his technology. An agent observing three incumbents, mimics the majority. An agent observing nothing chooses a technology based on his own idiosyncratic preferences.
One can show that this learning rule is a Nash equilibrium. Unique equilibrium if agents are sufficiently impatient.
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Introduction Model Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Example: Competing Technologies (Long run Behavior)
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 Main Results Extensions and Discussion
Motivation Research Question Brief Summary of Results Motivating Example
Example 2: Random Matching & Indirect Reciprocity Agents are randomly matched to play the Prisoner’s Dilemma. Learning rule: Agents play uniformly when they observe no past actions of the partner. When an agent observes the partner’s past actions, she plays the most frequently-observed action in her sample. The rule might be consistent with reciprocal preferences.
Same learning dynamics as in the previous example... Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Environment Learning Rule and Dynamics
Population State
Infinite population (continuum of agents with mass one). Time is discrete: 1, 2, 3, 4, 5, .... Each agent chooses action a ∈ A. Population state γ ∈ ∆ (A): aggregate distribution of actions.
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Introduction Model Main Results Extensions and Discussion
Environment Learning Rule and Dynamics
New Agents and Samples
At each period β ∈ (0, 1) 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).
Each new agent observes a sample with a random size l . ν - the distribution of the sample size l . The sampled actions are iid.
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Introduction Model Main Results Extensions and Discussion
Environment Learning Rule and Dynamics
Environment & Mean Sample Size An environment is a tuple E = (A, β, ν) describing: A - finite set of actions, β - fraction of new agents, ν distribution of sample size. Let µl denote the mean sample size, i.e., the expected number of actions observed by a random new agent in the population:
µl =
X
ν (l) · l.
l
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Introduction Model Main Results Extensions and Discussion
Environment Learning Rule and Dynamics
Learning Rule (Stationary) learning rule σ : M → ∆ (A) describes the behavior of a new agent who observes sample m ∈ M. E.g., taking the frequently-observed action in the motivating example.
A learning process is a pair consisting of an environment and a learning rule: P = (E , σ) .
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Introduction Model Main Results Extensions and Discussion
Environment Learning Rule and Dynamics
Population dynamics
An initial state & a learning process determine a new state. Let fP : Γ → Γ denote the mapping between states induced by a single step of the learning process P. We say that γ ∗ is a steady state if fP (γ ∗ ) = γ ∗ . For each t > 1, let fPt (ˆ γ ) denote the state induced after t steps of the learning process P, given an initial state γˆ .
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Introduction Model Main Results Extensions and Discussion
Environment Learning Rule and Dynamics
Local Stability and Global Attractors
γ ∗ is locally stable if a population beginning near γ ∗ remains close to γ ∗ , and eventually converges to γ ∗ . (Formally, ∀ > 0 there exists δ > 0 s.t. kˆ γ − γ ∗ k < δ implies: (1)
fPt (ˆγ ) − fPt (γ ∗ ) < ∀t ,(2) limt−→∞ fPt (ˆγ ) = γ ∗ .)
γ ∗ is an (almost-)global attractor, if the population converges to γ ∗ from any (interior) initial state. (formally, if limt−→∞ fPt (ˆ γ ) = γ ∗ for all γˆ (totally-mixed γˆ ).)
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Upper Bound on the Distance Between New States Theorem Distance between new populations states is at most 1 − β + β · µl times the distance between the old population states: kfP (γ) − fP (γ 0 )k1 ≤ (1 − β + β · µl ) · kγ − γ 0 k1 (strict inequality if agents may observe more than one action.)
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Upper Bound on the Distance Between New States Theorem Distance between new populations states is at most 1 − β + β · µl times the distance between the old population states: kfP (γ) − fP (γ 0 )k1 ≤ (1 − β + β · µl ) · kγ − γ 0 k1 (strict inequality if agents may observe more than one action.) Corollary µl ≤ 1 and ν (1) < 1 ⇒ fP is a weak contraction mapping (i.e., kfP (γ) − fP (γ 0 )k1 < kγ − γ 0 k1 ∀γ 6= γ 0 ) ⇒ fP admits a global attractor. Yuval Heller & Erik Mohlin
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Upper Bound – Sketch of Proof Sketch of Proof. Distance between distributions of actions of new agents ≤ Distance between distribution of samples ≤ Mean sample size * distance between old population states.
Upper Bound – Sketch of Proof Sketch of Proof. Distance between distributions of actions of new agents ≤ Distance between distribution of samples ≤ Mean sample size * distance between old population states. ⇒ k(fP (γ)) − (fP (γ 0 ))k1 ≤
Lemma 1
Def. of L1 Norm
P β · l ν (l) · ψl,γ − ψl,γ 0 1 + (1 − β) · kγ − γ 0 k1 ≤ (< if l > 1) P 0 0
β·
l
(β · (
ν (l) · l · kγ − γ k1 + (1 − β) · kγ − γ k1 = l∈N ν (l) · l) + (1 − β)) · kγ − γ
P
0k
=
(β · µl + 1 − β) · kγ − γ 0 k , where ψl,γ is the distribution of samples of length l, given γ.
Lem. 3
Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Characterizing Environments with Multiple Steady States
Theorem Let E be an environment. The following conditions are equivalent: 1
µl > 1 or ν (1) = 1.
2
There exists a learning rule σ, s.t. the learning process (E , σ) admits multiple steady states.
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Multiple Steady States: Sketch of Proof (1)
If agents always observe a single action (i.e., ν (1) = 1): Learning rule: each agent plays the observed action. Each initial state is a steady state. Consistent with best replying in the example of competing technologies.
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Multiple Steady States: Sketch of Proof (2) If µl > 1, then the learning rule is that each agent plays 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 play a, the rest play b.
Yuval Heller & Erik Mohlin
Lemma 4
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Multiple Steady States: Sketch of Proof (2) If µl > 1, then the learning rule is that each agent plays 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 play a, the rest play b.
Lemma 4
Remark This learning rule might be consistent with best-replying, e.g.,: Initial state is either no one plays a or 90% play a. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
Multiple Locally stable States
In the construction used in the previous result, only one of the steady states is locally stable. Moreover, the state x > 0 is an almost-global attractor.
What is the minimal mean sample size to imply that there is a rule with multiple locally stable states?
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
µl > 2 ⇒Multiple Locally stable States
Theorem Let E be an environment satisfying µl > 2. There exists rule σ, s.t. the process (E , σ) admits multiple locally stable states.
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
µl > 2 ⇒Multiple Locally stable States
Theorem Let E be an environment satisfying µl > 2. There exists rule σ, s.t. the process (E , σ) admits multiple locally stable states. Sketch of Proof (1): The Learning Rule. Observing action a twice or more ⇒ play a . Never observing action a ⇒ play b. Observing action a once ⇒ play q·a and (1 − q) ·b, where q < µ1 and
1
µl l Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
µl > 2 ⇒Multiple Locally stable States Sketch of Proof (2). Assume that a is played with frequency x << 1 (“state x ”). Share of new agents who play a is f (x ) = q · µl · x + (1 − 2 · q) · O x 2 .
q<
1 µl
⇒ state x very close to zero converges to zero
(because q · µl · x + O x 2 < x ).
1 µl
− q << 1 ⇒ state in which a few more agents play a
converges to more agents playing a (because 1 − 2 · q > 0). Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States We present two families of environments, each with sample sizes 1 ≤ µl ≤ 2: First family: Each learning rule admits at most one locally stable state. Second family: There is a learning rule with two locally stable states.
Conclusion: Some (but not all) environments with 1 ≤ µl ≤ 2 admit a learning rule with multiple locally stable states. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
l ≤ 2 ⇒ At Most One Locally stable State Theorem Let E = ({a, b} , β, ν) be an environment satisfying supp (ν) ⊆ {0, 1, 2}. Then for any rule σ, process (E , σ) admits at most 1 locally stable state.
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
l ≤ 2 ⇒ At Most One Locally stable State Theorem Let E = ({a, b} , β, ν) be an environment satisfying supp (ν) ⊆ {0, 1, 2}. Then for any rule σ, process (E , σ) admits at most 1 locally stable state.
Sketch of Proof. State is the frequency x ∈ [0, 1] of agents playing a. Maximal sample size is 2 ⇒ fσ (x ) is a polynomial of degree two. ⇒ There are at most two steady states solving fσ (x ) = x . Simple geometric arguments: At most one of these steady states can be locally stable (see figures in the next slide). Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
l ≤ 2 ⇒ At Most One Locally stable State
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
l∈ {1, 3}⇒ “Follow Majority” has 2 Locally Stable States Theorem Let E = ({a, b} , β, ν) be an environment. Assume that: ν (1) < 1 and ν (1) + ν (3) = 1. Then there is a rule σ ∗ , such that environment (E , σ ∗ ) admits two locally stable states.
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Introduction Model Main Results Extensions and Discussion
Upper Bound µl > 1 ⇔ Multiple Steady States µl > 2 ⇒ Multiple Locally stable States 1 ≤ µl ≤ 2: Sometimes Multiple Locally stable States
l∈ {1, 3}⇒ “Follow Majority” has 2 Locally Stable States Theorem Let E = ({a, b} , β, ν) be an environment. Assume that: ν (1) < 1 and ν (1) + ν (3) = 1. Then there is a rule σ ∗ , such that environment (E , σ ∗ ) admits two locally stable states. Sketch of Proof. Learning rule: Each agents plays the frequently-observed action in his sample (as in the motivating example). Two locally stable states: everyone playing a and everyone playing b (and an unstable steady state: 0.5 · a + 0.5 · b). Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
Responsiveness and effective Sample Size
In the paper we present an upper bound to how much a learning rule is responsive to a change in the sample. We use this bound to define an effective sample size, which is (weakly) smaller than the mean sample size. We use this effective sample size, to achieve a (weakly) stronger result on learning processes that admit global attractors. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
Heterogeneous Populations
In many applications the population might be heterogeneous (i.e., the population includes various groups that differ in their sampling procedures and learning rules). E.g., Ellison & Fudenberg (93), Young (93), Munshi (04).
In the paper, we formally extend our model and results to deal with heterogeneous populations.
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
Non-Stationary Environments and Common Shocks In the paper we extend our model and results to deal with time-dependent learning rules, and we characterize when a non-stationary environment globally converges to a time-dependent sequence of states that is independent of the initial state. We further extend the model to deal with stochastic shocks that influence the learning rules of all agents (on the aggregate level), and we characterize when the initial population state may have a lasting effect in such environments. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
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 stationary strategies used by the players is sufficient to uniquely determine the steady state? Our main results show that this is true if: (1) µl < 1, or (2) l ≤ 2 and |A| = 2. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
Related Literature Existing literature focuses on a specific rule - myopic best reply. Arthur (1989) (& Arthur, 1994; Kaniovski & Young, 1995): Social learning is path dependent when technologies have increasing returns.
Young (1993) & Kandori et al. (1993) stability in the long run is independent of the initial conditions. Sandholm (2001) (& Oyama et al., 2015): agents observe k actions and the game admits a
1 k -dominant
action a∗ ⇒ global convergence to a∗ .
Our model differs from the existing literature by studying general environments with arbitrary learning rules (and arbitrary payoffs). Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
Conclusion Mean sample size (µl ) Is there a rule with multiple steady states? Is there a rule with multiple
µl ≤ 1
µl ∈ (1, 2]
µl > 2 Yes
No (global convergence)
Sometimes
Yes
locally stable states? Extensions: Heterogeneous populations, non-stationary environments, and common random shocks. Yuval Heller & Erik Mohlin
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Introduction Model Main Results Extensions and Discussion
Extensions Discussion & Related Literature Conclusion Backup Slides
Backup Slides
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Introduction Model Main Results Extensions and Discussion
L1 -Norm - Definitions
Extensions Discussion & Related Literature Conclusion Backup Slides
Back
Let the L1 -distance between ψl,γ , ψl,γ 0 ∈ ∆ Al , γ, γ 0 ∈ ∆ (A):
ψl,γ − ψl,γ 0 = P m∈Al ψl,γ (m) − ψl,γ 0 (m) , 1 P 0 0
kγ − γ k1 =
a∈A |γ (a) − γ
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(a)| .
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Proof of Lemma 1
Back
P
k(fP (γ)) − (fP (γ 0 ))k1 =
a∈A
|(fP (γ)) (a) − (fP (γ 0 )) (a)| =
P P β · ν (l) m∈Al ψl,γ (m) · σm + (1 − β) · γ (a) l P P − β · l ν (l) · m∈Al ψl,γ 0 (m) · σm + (1 − β) · γ 0 (a) = P P P 0 β· ν (l) · l ψl,γ (m) − ψl,γ 0 (m) · σm (a) + (1 − β) · (γ (a) − γ (a)) ≤
P
a∈A
a∈A
l
m∈A
(4 Ineq.)
P a∈A
β·
P l
P a∈A
β·
P
ν (l) ·
l
P
ν (l) ·
P a∈A
m∈Al
P
|γ (a) − γ 0 (a)|≤
m∈Al
ψl,γ (m) − ψl,γ 0 (m) · σm (a) + (1 − β) · |γ (a) − γ 0 (a)| =
ψl,γ (m) − ψl,γ 0 (m) · σm (a) + (1 − β) ·
Lemma 2
β·
P l
ν (l) · ψl,γ − ψl,γ 0 + (1 − β) · γ − γ00 . 1
1
Introduction Model Main Results Extensions and Discussion
Proof of Lemma 2
(4 ≤)
Extensions Discussion & Related Literature Conclusion Backup Slides
Back
X X ψl,γ (m) − ψl,γ 0 (m) · σm (a) a∈A m∈Al X X ψl,γ (m) − ψl,γ 0 (m) · σm (a) ≤ a∈A m∈Al
=
X X ψl,γ (m) − ψl,γ 0 (m) · σm (a) m∈Al
a∈A
X
ψl,γ (m) − ψl,γ 0 (m) · 1 = ψl,γ − ψl,γ 0 . = 1 m∈Al
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Proof of Lemma 3
Back
− → − → =
ψl,γ − ψl,γ 0 = P→ − a ∈Al ψl,γ a − ψl,γ 0 a 1 P Q Q 0 γ (a ) − γ (a ) → − = telescoping series i i a 1≤i≤l 1≤i≤l P Q Q P 0 0 → − a ∈Al 1≤i≤l γ (ai ) − γ (ai ) · i 1)
P
Q Q 0 γ (ai ) − γ 0 (ai ) · γ a · γ a = j j i
→ − a ∈Al
(ai ,...,ai−1 )∈Ai−1
1≤j
γ 0 aj
γ (ai ) − γ 0 (ai ) · 1 · 1 = P
γ − γ 0 = 1≤i≤l 1
l · γ − γ 0 1 ≤ l · γ − γ 0 . =
P
1≤i≤l
P
ai ∈A
Introduction Model Main Results Extensions and Discussion
Telescoping series
Extensions Discussion & Related Literature Conclusion Backup Slides
Back
Y
ai −
1≤i≤l
Y
bi =
1≤i≤l
(a1 · ... · an − b1 · a2 · ... · an ) + (b1 · a2 · ... · an + b1 · b2 · a3 · ... · an ) − b1 · b2 · a3 · ... · an + ... + b1 · ... · bn = (a1 − b1 ) · a2 · ... · an + (a2 − b2 ) · a3 · ... · an · b1 + (a3 − b3 ) · a4 · ... · an · b1 · b2 ... + (an − bn ) · b2 · ... · bn =
=
X 1≤i≤l
Yuval Heller & Erik Mohlin
(ai − bi ) ·
Y i
aj ·
Y
bj .
1≤j
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Introduction Model Main Results Extensions and Discussion
Proof of Lemma 4
Extensions Discussion & Related Literature Conclusion Backup Slides
Back
A share of x ∗ playing a∗ is a steady state iff: P P l ∗) = ν (l) · Pr (sampling a ν (l) · 1 − (1 − x ) ≡ g (x ) = x θ l l g (x ) is continuous & increasing, g (1) ≤ 1.. x << 1 ⇒ 1 − (1 − x )l can be (Taylor-)approximated by 1 − (1 − x )l = 1 − 1 − l · x + O x 2 = l · x + O x 2 ⇒ g (x ) = µl · x + O x 2 > x ∗
⇒ ∃0 < x ∗ ≤ 1 s.t. g (x ∗ ) = x ∗ ⇒ γ x is a steady state. Yuval Heller & Erik Mohlin
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