Dynamic Adoption, Information Lag,and Coordination Problem

in the Presence of Network Externalities

In-Uck Park (joint with Jack Ochs)June 2006 (Work in Progress)

1. Introduction

Network commodities exhibit positive adoption externalities.

• Examples: telecommunication, etc, diffusion of innovations andstandards.

• Coordination is well-known issue: Pareto-ranked equilibria, in-cluding the null one and the full/maximal one.

• We study the effect of a dynamic adoption process on resolvingthe coordination problem when agents have private types.

Main Results

• Coordination problem may not be resolved by dynamic adop-tion process so long as follow-up adoptions take place after atime lag.

• It is resolved if there is no time lag, in the sense that there is aunique symmetric equilibrium.

• Asymmetric equilibrium also disappears if the population sizeis perceived as random (in a sense to be made precise).

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Uniqueness of equilibrium in pure coordination games Simultaneous Sequential Decision Decision Insurance schemes Standard backward Complete implement the maximum induction guarantees Info equilibrium uniquely unique equilibrium. at zero ex post cost. Farrell-Saloner (1985) Dybvig-Spatt (1983) Gale (1995) Dixit (2003) Simple inducement scheme “Stochastic” backward Incomplete exists that warrants unique induction establishes Info symmetric equilibrium. unique symmetric equilibrium. Park (2004) Farrell-Saloner (1985), Current paper furthers The investigation.

2. Discrete-time Model

Game Tree:• There are N +1 ex ante identical agents, indexed by i ∈ I =1, · · · , N+1.

• Each i has a private type t independently drawn from a commondensity function f : <+ → <+ (continuous and bounded).

• At the beginning of each period k = 1, · · ·, the number nk−1

of agents who already joined/adopted is publicly known; Basedon history hk := (n1, · · · , nk−1), the remaining agents simulta-neously decide whether to join or not.

Stage Utility ut(ν) ∈ < of an adopter in period k is determinedby his type t and the current network size, measured by the numberν(= nk − 1) of other adopters (i.e., not counting himself); ut(ν) isstrictly increasing in ν = 0, · · · , N .

A non-adopter’s utility is normalized to uφ = 0. Each agentmaximizes the expected δ-discounted average of utility stream witha discount factor δ < 1:

(1− δ)∞∑

k=1

δk−1 · (stage utility of period k) (1)

Type t ≥ 0 measures how reluctant an agent is to join the network:ut(ν) is strictly decreasing and continuous in t, and

• u0(0) = 0 : the “best” type is indifferent between adoptingalone and not adopting (this is not critical);

• There is t such that ut(N) = 0, which is unique. (Wlog, t = 1.)

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Strategy: A remaining agent i’s period-k strategy given a historyhk = (n1, · · · , nk−1), is an integrable function

ai(·|hk) : <+ → [0, 1]

where ai(t|hk) is the probability that the agent i joins in period k

when his type is t. It is a cutoff strategy at (a cutoff level) t ≥ 0, ifai(t|hk) = 1 for all t < t and ai(t|hk) = 0 for all t > t.

An agent i’s strategy is a collection ai(·|hk) for all possiblehk and k = 1, · · · ,K, which we denote by ai as shorthand. It is acutoff strategy if each ai(·|hk) is.

Equilibrium: A strategy profile (ai)i∈I is a (perfect Bayesian) equi-librium if each agent i’s period-k strategy after each hk is a bestresponse to (aj)j 6=i conditional on hk.

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Benchmark: Static case, i.e., when δ = 0.

• An equilibrium is characterized by the common cutoff level t∗

that satisfies:

N∑ν=0

(N

ν

)F (t∗)ν(1− F (t∗))N−ν · ut∗(ν) = 0. (2)

• t∗ = 0 is an equilibrium.

• There is a maximum equilibrium cutoff level t∗ < t.

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3. Unique Symmetric Equilibrium

• Construct a unique symmetric equilibrium for δ = 1: it is acutoff equilibrium and the cutoff level in each period dependsonly on nk−1.¦ When δ = 1, only the final network size matters, not the

path to it.¦ However, inessential multiplicity b/c completion time does

not matter.• Hence, assume a stopping rule: if no one joins in some period,

the joining process stops (i.e., no further adoption is allowed).• Then, show that there exists a threshold δ∗ < 1 such that the

same argument can be extended to all δ > δ∗ to establish uniquesymmetric equilibrium: it is a cutoff equilibrium, however, thecutoff level in each period depends on the full adoption historyup to then.

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We construct symmetric equilibrium (ai)i∈I for δ = 1 by an induc-tion argument.

• We refer to the number of remaining agents at the beginning ofperiod k, s = N + 1− nk, as the state.

Step 1: Suppose s = 1, i.e., only one agent remains in some periodk. Clearly he adopts iff his type ≤ t, i.e., a cutoff strategy at τ1 ≡ t.

Step 2: Suppose s = 2 in period k; let τ2 be this period’s cutofflevel. Clearly τ2 < τ1. Consider a remaining agent i of type τ2.

• Waiting leads to the same outcome as adopting if the otheragent were to adopt in the current period, because agent i can“catch up later” when the other adopts now.

• Hence, focus the contingency that the other agent not adopt:If joins ⇒ ut(N) · F |τ2

(τ1) + ut(N − 1) · (1− F |τ2(τ1))

If waits ⇒ u∅ = 0.• The former decrease in τ2, hence equilibrium τ2 value is unique.

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0 t2 1=t1 these types these types will join next pd will not join

Step 3: Suppose s = 3 in period k; let t3 be this period’s cutoff level. Clearly t3 < t2. The uniqueness is less clear—see diagram below. We prove that uniqueness is still obtained. Theorem 1. If d=1, there exists a unique symmetric equilibrium of G. In this equilibrium, the remaining agents’ strategy is a cutoff strategy that depends only on the state s, denoted by the cutoff level ts, and 0 < tN+1 < tN < … < t1 = t . Uniqueness when d<1 • For d<1, the path to the final network matters and catching up

may not be possible sometimes. • The effects of these discrepancies are negligible for d close to

1. flThe expected utilities from waiting and joining, as functions of t, are close to those when d=1. fl The solution set changes continuously near d=1.

Theorem 2. Suppose i) )(νtu& , the derivative of )(νtu with respect to t, exists for all ),0( tt ∈ and N,,0L=ν , and ii) there is q>0 such that θν >|)(| tu& for all ),0( tt ∈ and N,,0L=ν . There is d*<1 such that if d>d* there is a unique symmetric equilibrium of G. Furthermore, this equilibrium is a cutoff equilibrium and converges to the equilibrium described in Theorem 1 as dØ1.

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When the stopping rule is dropped

• If no one joins in some period, then an equilibrium of the continuation game is played from the next period.

Vacation Equilibrium

• After a certain history, the adoption process freezes for one (or more) period and then resumes.

• E.g, for 3 agents, if only one agent enters in the first period, then the next period is “vacation”; and from period 3 the remaining two agents resume the continuation game. This affects the cutoff levels if d<1.

More Equilibria

• E.g, for 2 agents, two equilibria as below:

0 t2 1=t1

0 t0 t2 1=t1 In the 2nd equil, the cutoff level is t0 in the initial period;

If no one enters then the cutoff level is t2 in the next period. As d→1, t2 returns to the original level and t0→0.

These variations arise because of the (potential) utility loss of the adopter due to a time lag until any follow-up adoptions.

4. Continuous-time Model

There is no natural notion of “first time after k.” We follow theextensive-form formalization by Simon–Stinchcombe (1989, Ecta).

• Time is k ∈ <+.• Simultaneous adoptions by agents at time k ∈ <+, is repre-

sented by a jump (k, s) where s is the resulting network size.• Multiple jumps may take place consecutively at the same in-

stant of time k.• A history is a string of jumps that took place in sequel: h =

((k1, s1), · · · , (k`, s`)).• A decision node is a pair (h, k) where k ≥ k`.• At each decision node the remaining agents decide whether to

enter or not.

Theorem 3. For all δ < 1 there exists a unique symmetric equi-

librium. In this equilibrium, adoptions take place in the same order

as in the unique symmetric equilibrium of the discrete-time model

with δ = 1, consecutively yet all at the same time k = 0.

For this result, important are:

• Follow-up adoptions may take place without delay, so that soleadoption is not costly as long as more adoptions follow.

• The “last chance of adoption” does not exist.

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“Subscribe prior to service” does not work

• All adoption must happen at discrete points of interval ε >0 during )1,0[∈k and service starts from k>1.

• Then there is the last adoption time 1<k . • Everyone playing a static equilibrium at k may be an equil.

Example: 3 agents

Sequential equil: 0 t3=.2 t2=.7 1=t1

Static equil: 0 t*=.95 N.B. Types < .2 can enforce the sequential equil by adopting in

period 1, however they prefer the static equil.

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5. Asymmetric Equilibria • Asymmetric equilibrium may exist in some environments.

Example: 2 agents

0 t2 1=t1

0 tS tR 1=t1

• However, for any two agents, their respective sets of “all other agents” that they best-respond to become arbitrarily close as the number of agents, N, gets large, so all asymmetric equilibria converge to the unique symmetric equilibrium as ∞→N .

Uncertain population

• Agents perceive N as a random variable. • Then, it is natural that each agent treats other agents

symmetrically, hence symmetric equilibrium (Poisson Games-Myerson, 1998).

• We try to dismiss asymmetric equilibrium in a more general setting from the premise that each agent treats other agents symmetrically.

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A description of a situation in which distinct strategies A, B, C,…, are used by different agents. World of possibilities Ω : N=5: (2A,2B,1C), (2A,1B,2C), (1A,2B,2C), … N=4: (2A,2B), (2A,1B,1C), (1A,2B,1C), (1A,1B,2C), (2B,2C) N=3: (2A,1B), (1A,1B,1C), (2B,1C) There is a distribution function d on Ω. Each agent has a transition probability π. Consistency: Fix an arbitrary agent a who uses a strategy, say A.

• Conditional on a being in a scenario of a particular N, calculate the probabilities that a typical other agent uses A, B,… These do not depend on N, and are equal to those of an additional agent when a “transited” from a scenario of N to one of N+1.

• The last set of probabilities is the same across agents. Theorem 4: If (Ω, d) satisfies Consistency, then all agents use identical strategies. Theorem 5: If the population size in uncertain, there is a unique equilibrium. This equilibrium is a symmetric cutoff equilibrium.

World of possibilities (Ω, d)• Set of possible population sizes is N = N0, N0 + 1, · · · , NM.• There is a set S(N) of scenarios (n(A), n(B), · · · , n(Z)) for

each N ∈ N , where∑Z

i=A n(i) = N .• Any scenario of S(N + 1) can be obtained from a scenario of

S(N) by adding one agent.• There is a distribution function d on Ω = ∪NS(N).

Consistency - Consider an agent a who uses a certain strategy, sayA and fix any two strategies, say X and Y .• Let pa(X|N) be the probability that a randomly chosen agent

other than a uses X, conditional on the agent a is in a scenarioof S(N). Define pa(Y |N) analogously.

• Agent a has a transition probability πa in mind from each sce-nario in S(N) to the scenarios in S(N + 1) that are obtainableby adding an agent, which is consistent with (Ω, d).

• Let ra(X|N) be the probability that the additional agent usesX, conditional on the agent a has “transited” from a scenarioof S(N) to a scenario of S(N + 1) according to πa. Definera(Y |N) analogously.

• For an agent b who uses B, define pb(X|N), pb(Y |N), rb(X|N)and rb(Y |N) analogously.

• Consistency requires for i = a, b and all X, Y ,

pi(X|N)pi(Y |N)

=pi(X|N + 1)pi(Y |N + 1)

=ri(X|N)ri(Y |N)

, and

ra(X|N)ra(Y |N)

=rb(X|N)rb(Y |N)

.

(C)

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